https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02LIAG:Kapitola7&feed=atom&action=history 02LIAG:Kapitola7 - Historie editací 2024-03-29T11:49:10Z Historie editací této stránky MediaWiki 1.25.2 https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02LIAG:Kapitola7&diff=6407&oldid=prev Hazalmat v 5. 8. 2016, 00:06 2016-08-05T00:06:34Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 5. 8. 2016, 00:06</td> </tr><tr><td colspan="2" class="diff-lineno" id="L474" >Řádka 474:</td> <td colspan="2" class="diff-lineno">Řádka 474:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> Dále libovolné $V$ nad $\C$ lze rozložit jako</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> Dále libovolné $V$ nad $\C$ lze rozložit jako</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \begin{align*}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \begin{align*}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> V=\bigoplus_{\lambda \in \sigma(<del class="diffchange diffchange-inline">A</del>)}\lim_{n \to +\infty}\ker(<del class="diffchange diffchange-inline">A </del>- \lambda\mathbb{1})^n,\qquad \forall <del class="diffchange diffchange-inline">A </del>\in \gl(V).</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> V=\bigoplus_{\lambda \in \sigma(<ins class="diffchange diffchange-inline">X</ins>)}\lim_{n \to +\infty}\ker(<ins class="diffchange diffchange-inline">X </ins>- \lambda\mathbb{1})^n,\qquad \forall <ins class="diffchange diffchange-inline">X </ins>\in \gl(V).</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{align*}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{align*}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> Indukcí ukážeme:</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> Indukcí ukážeme:</div></td></tr> </table> Hazalmat https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02LIAG:Kapitola7&diff=6402&oldid=prev Hazalmat v 4. 8. 2016, 22:21 2016-08-04T22:21:40Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 4. 8. 2016, 22:21</td> </tr><tr><td colspan="2" class="diff-lineno" id="L99" >Řádka 99:</td> <td colspan="2" class="diff-lineno">Řádka 99:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \begin{align*}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \begin{align*}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \left[ \mrm{tr}(n),\mrm{tr}(n) \right] = \mrm{str}(n),\quad \left[ \mrm{tr}(n), \mrm{str}(n) \right] = \mrm{str}(n)</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \left[ \mrm{tr}(n),\mrm{tr}(n) \right] = \mrm{str}(n),\quad \left[ \mrm{tr}(n), \mrm{str}(n) \right] = \mrm{str}(n)</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> \end{align*} $\<del class="diffchange diffchange-inline">rimpl</del>$ je řešitelná, ale není nilpotentí.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> \end{align*} &#160;</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"> </ins>$\<ins class="diffchange diffchange-inline">Rightarrow\quad</ins>$je řešitelná, ale není nilpotentí.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> } </div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> } </div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Def{</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Def{</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L115" >Řádka 115:</td> <td colspan="2" class="diff-lineno">Řádka 116:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> přičemž $\s$ je určena až na izomorfii. $\s$ nebo $\rr$ může být rovna $0$. Bez důkazu.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> přičemž $\s$ je určena až na izomorfii. $\s$ nebo $\rr$ může být rovna $0$. Bez důkazu.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\subsection{Vlastnosti ideálů (cvičení)}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\subsection{Vlastnosti ideálů (cvičení)}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Vet{</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Vet{</div></td></tr> </table> Hazalmat https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02LIAG:Kapitola7&diff=6401&oldid=prev Hazalmat v 4. 8. 2016, 22:18 2016-08-04T22:18:12Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 4. 8. 2016, 22:18</td> </tr><tr><td colspan="2" class="diff-lineno" id="L467" >Řádka 467:</td> <td colspan="2" class="diff-lineno">Řádka 467:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> $\ad_\g = \left\{ \ad_X \middle| X \in \g \right\}$ je tvořena nilpotentními operátory$\quad\xRightarrow{Lemma\ \ref{Lemma4:Lie-Engel}}\quad \ad_\g$ je nilpotentní maticová algebra$\rimpl \g$ je nilpotentní. Opačná implikace plyne z poznámky \zref{nilpotentni poznamka}.<del class="diffchange diffchange-inline">\\</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> $\ad_\g = \left\{ \ad_X \middle| X \in \g \right\}$ je tvořena nilpotentními operátory$\quad\xRightarrow{Lemma\ \ref{Lemma4:Lie-Engel}}\quad \ad_\g$ je nilpotentní maticová algebra$\rimpl \g$ je nilpotentní. Opačná implikace plyne z poznámky \zref{nilpotentni poznamka}.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> Dále libovolné $V$ nad $\C$ lze rozložit jako</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> Dále libovolné $V$ nad $\C$ lze rozložit jako</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L495" >Řádka 495:</td> <td colspan="2" class="diff-lineno">Řádka 495:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> Indukcí na $\dim \g$: $\dim \g =1$ zřejmé \\</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> Indukcí na $\dim \g$: $\dim \g =1$ zřejmé<ins class="diffchange diffchange-inline">. </ins>\\</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> $\dim \g = k-1 \to k$: $\g$ řešitelná, $\dim \g = k \rimpl \g^{(1)} = [\g,\g] \subsetneqq \g$, vezmeme $\h$ podprostor $\g,\ \g^{(1)} \subset \h,\ \dim \h = k-1 \rimpl \h$ je ideál, protože $[\h,\g] \subset \g^{(1)} \subset \h$ a protože $\h$ je řešitelný splňuje indukční předpoklad$\rimpl \exists v_0 \in V,\ v \neq 0,\ Xv_0 = \lambda_0(X)v_0,\ \forall X \in \h$.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> $\dim \g = k-1 \to k$: $\g$ řešitelná, $\dim \g = k \rimpl \g^{(1)} = [\g,\g] \subsetneqq \g$, vezmeme $\h$ podprostor $\g,\ \g^{(1)} \subset \h,\ \dim \h = k-1 \rimpl \h$ je ideál, protože $[\h,\g] \subset \g^{(1)} \subset \h$ a protože $\h$ je řešitelný splňuje indukční předpoklad$\rimpl \exists v_0 \in V,\ v \neq 0,\ Xv_0 = \lambda_0(X)v_0,\ \forall X \in \h$.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> Vezmeme libovolné $Z \in \g \setminus \h$, tedy $\g = \h + \mrm{span}\{ Z \}$, a definujeme $v_{j+1} = Zv_j,\ j \in \N_0,\ W = \mrm{span}\{ v_j \}_{j=0}^{+\infty}$. Platí ale $\dim W \leq \dim V &lt; +\infty \rimpl \exists p \in \N,\ W = \mrm{span}\{ v_j \}_{j=0}^p,\ ZW \subset W$. Pro libovolné $X \in \h$ platí:</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> Vezmeme libovolné $Z \in \g \setminus \h$, tedy $\g = \h + \mrm{span}\{ Z \}$, a definujeme $v_{j+1} = Zv_j,\ j \in \N_0,\ W = \mrm{span}\{ v_j \}_{j=0}^{+\infty}$. Platí ale $\dim W \leq \dim V &lt; +\infty \rimpl \exists p \in \N,\ W = \mrm{span}\{ v_j \}_{j=0}^p,\ ZW \subset W$. Pro libovolné $X \in \h$ platí:</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L516" >Řádka 516:</td> <td colspan="2" class="diff-lineno">Řádka 516:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> } </div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> } </div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> $\g$ tvoří horní trojúhelníkové matice$\rimpl \g$ řešitelná. <del class="diffchange diffchange-inline">\\</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> $\g$ tvoří horní trojúhelníkové matice$\rimpl \g$ řešitelná. &#160;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> Naopak, mějme řešitelnou algebru $\g\subset \gl(V),\ V$ nad $\C \quad \xRightarrow{Lemma\ \ref{Lemma5:Lie-Engel} }\quad \exists v_1 \in V,\ v_1 \neq 0,\ \lambda_1 \in \g^*,\ \forall X \in \g,\ Xv_1 = \lambda_1(X)v_1$. Definujeme $V_1 := \mrm{span}\{ v_1 \}$ a reprezentaci $\g$ na $V/V_1$:</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> Naopak, mějme řešitelnou algebru $\g\subset \gl(V),\ V$ nad $\C \quad \xRightarrow{Lemma\ \ref{Lemma5:Lie-Engel} }\quad \exists v_1 \in V,\ v_1 \neq 0,\ \lambda_1 \in \g^*,\ \forall X \in \g,\ Xv_1 = \lambda_1(X)v_1$. Definujeme $V_1 := \mrm{span}\{ v_1 \}$ a reprezentaci $\g$ na $V/V_1$:</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L541" >Řádka 541:</td> <td colspan="2" class="diff-lineno">Řádka 541:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \left( \g_\C \right)^k = \left(\g^k\right)_\C,\qquad \left( \g_\C \right)^{(k)} = \left(\g^{(k)}\right)_\C</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \left( \g_\C \right)^k = \left(\g^k\right)_\C,\qquad \left( \g_\C \right)^{(k)} = \left(\g^{(k)}\right)_\C</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{align*}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{align*}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> $\Rightarrow\quad$platí proto: $\quad\g_\C$ je řešitelná (resp. nilpotentí)$\quad\Leftrightarrow\quad \g$ je řešitelná (resp. nilpotentní). <del class="diffchange diffchange-inline">\\</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> $\Rightarrow\quad$platí proto: $\quad\g_\C$ je řešitelná (resp. nilpotentí)$\quad\Leftrightarrow\quad \g$ je řešitelná (resp. nilpotentní). &#160;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> Stačí tedy ukázat platnost pro $V$ nad $\C$: \\</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> Stačí tedy ukázat platnost pro $V$ nad $\C$: \\</div></td></tr> </table> Hazalmat https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02LIAG:Kapitola7&diff=6400&oldid=prev Hazalmat v 4. 8. 2016, 22:12 2016-08-04T22:12:01Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 4. 8. 2016, 22:12</td> </tr><tr><td colspan="2" class="diff-lineno" id="L435" >Řádka 435:</td> <td colspan="2" class="diff-lineno">Řádka 435:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \exists Z \in \g: Z + \h \in \g / \h,\ Z + \h \neq \h,\ \phi(X)(Z + \h) = \h,\ \forall X \in \h \rimpl Z \notin \h,\ [X,Z] \in \h,\ \forall X \in \h</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \exists Z \in \g: Z + \h \in \g / \h,\ Z + \h \neq \h,\ \phi(X)(Z + \h) = \h,\ \forall X \in \h \rimpl Z \notin \h,\ [X,Z] \in \h,\ \forall X \in \h</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{align*}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{align*}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> $\Rightarrow\quad \mrm{span}\{ Z \} + \h$ je podalgebra $\g$, její $\dim = \dim \h +1$ a z maximality $\h$ plyne, že je to celé $\g$. <del class="diffchange diffchange-inline">\\</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> $\Rightarrow\quad \mrm{span}\{ Z \} + \h$ je podalgebra $\g$, její $\dim = \dim \h +1$ a z maximality $\h$ plyne, že je to celé $\g$. &#160;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> Z indukcního předpokladu rovnež $\exists v \in V,\ \h v = 0$, tzn. $W := \bigcap_{X \in \h}\ker X \neq \{ 0 \}$ a $\h$ je ideál, neboť $[\g,\h] = [\mrm{span}\{ Z \} + \h, \h] \subset \h \quad\xRightarrow{Lemma\ \ref{Lemma2:Lie-Engel} }\quad W$ je invariantní podprostor $\g.$ A protože $Z$ je nilpotentní, je taky $\zuz{Z}{W}: W \to W$ nilpotentní, tedy</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> Z indukcního předpokladu rovnež $\exists v \in V,\ \h v = 0$, tzn. $W := \bigcap_{X \in \h}\ker X \neq \{ 0 \}$ a $\h$ je ideál, neboť $[\g,\h] = [\mrm{span}\{ Z \} + \h, \h] \subset \h \quad\xRightarrow{Lemma\ \ref{Lemma2:Lie-Engel} }\quad W$ je invariantní podprostor $\g.$ A protože $Z$ je nilpotentní, je taky $\zuz{Z}{W}: W \to W$ nilpotentní, tedy</div></td></tr> </table> Hazalmat https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02LIAG:Kapitola7&diff=6399&oldid=prev Hazalmat v 4. 8. 2016, 22:04 2016-08-04T22:04:30Z <p></p> <a href="https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02LIAG:Kapitola7&amp;diff=6399&amp;oldid=6398">Ukázat změny</a> Hazalmat https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02LIAG:Kapitola7&diff=6398&oldid=prev Hazalmat v 4. 8. 2016, 20:43 2016-08-04T20:43:32Z <p></p> <a href="https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02LIAG:Kapitola7&amp;diff=6398&amp;oldid=6376">Ukázat změny</a> Hazalmat https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02LIAG:Kapitola7&diff=6376&oldid=prev Hazalmat v 31. 7. 2016, 14:03 2016-07-31T14:03:21Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 31. 7. 2016, 14:03</td> </tr><tr><td colspan="2" class="diff-lineno" id="L101" >Řádka 101:</td> <td colspan="2" class="diff-lineno">Řádka 101:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> } </div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> } </div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Pzn{</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Pzn{</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> Nechť $\rr$ je radikál algebry $\g$. Uvažujme $\g / \rr$ a v ní abelovksý ideál $\h / \rr \subset \g / \rr$, tj. $\rr \subset \h \subset g,\ [\h,\h] \subset \rr \rimpl \h^{(1)} <del class="diffchange diffchange-inline">= </del>\rr$ a současně $\exists k,\ \rr^{(k)} = 0$, tj. $\rr$ je řešitelná$\rimpl \h^{(k+1)} \subset \rr^{(k)} = 0 \rimpl$podle předpokladu maximality $\rr$ je $\h = \rr \rimpl \g / \rr$ nemá netriviální abelovský ideál$\rimpl \g / \rr$ je poloprostá.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> Nechť $\rr$ je radikál algebry $\g$. Uvažujme $\g / \rr$ a v ní abelovksý ideál $\h / \rr \subset \g / \rr$, tj. $\rr \subset \h \subset g,\ [\h,\h] \subset \rr \rimpl \h^{(1)} <ins class="diffchange diffchange-inline">\subset </ins>\rr$ a současně $\exists k,\ \rr^{(k)} = 0$, tj. $\rr$ je řešitelná$\rimpl \h^{(k+1)} \subset \rr^{(k)} = 0 \rimpl$podle předpokladu maximality $\rr$ je $\h = \rr \rimpl \g / \rr$ nemá netriviální abelovský ideál$\rimpl \g / \rr$ je poloprostá.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> } </div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> } </div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Platí dokonce ješte silnější tvrzení: </div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Platí dokonce ješte silnější tvrzení: </div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L171" >Řádka 171:</td> <td colspan="2" class="diff-lineno">Řádka 171:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> Pro $\forall X \in g$ máme $X(f)(p) = \zuz{\td{}{s}}{s=0}\left( f\circ\phi_s^X \right)(p) = \zuz{\td{}{s}}{s=0}f(p\e^{sX}),\ \forall f \in \Cs^\infty(G)$ a zároveň $\e^{\phi_*(X)} = \phi\left( \e^X \right),\ \forall \phi$ homomorfismus.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> Pro $\forall X \in g$ máme $X(f)(p) = \zuz{\td{}{s}}{s=0}\left( f\circ\phi_s^X \right)(p) = \zuz{\td{}{s}}{s=0}f(p\e^{sX}),\ \forall f \in \Cs^\infty(G)$ a zároveň $\e^{\phi_*(X)} = \phi\left( \e^X \right),\ \forall \phi$ homomorfismus.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \begin{gather*}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \begin{gather*}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> \zuz{\ad_X(Y)f}{p} = \zuz{\td{}{t}}{t=0}\zuz{\Ad_{\e^{tX}}(Y)f}{p} = \zuz{\td{}{t}}{t=0}\zuz{\phi_{\e^{tX}*}(Y)f}{p} = \zuz{\td{}{s}}{s=0}\zuz{\td{}{t}}{t=0}\zuz{f\circ R_{\e^{s\phi_{\e^{tX}*<del class="diffchange diffchange-inline">}}</del>}(Y)}{p} = \\</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> \zuz{\ad_X(Y)f}{p} = \zuz{\td{}{t}}{t=0}\zuz{\Ad_{\e^{tX}}(Y)f}{p} = \zuz{\td{}{t}}{t=0}\zuz{\phi_{\e^{tX}*}(Y)f}{p} = \zuz{\td{}{s}}{s=0}\zuz{\td{}{t}}{t=0}\zuz{f\circ R_{\e^{s\phi_{\e^{tX}*}(Y)<ins class="diffchange diffchange-inline">}}</ins>}{p} = \\</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> = \zuz{\td{}{s}}{s=0}\zuz{\td{}{t}}{t=0}f\left( p\e^{s\phi_{\e^{tX}*}(Y)}\right) = \zuz{\td{}{s}}{s=0}\zuz{\td{}{t}}{t=0}f\left( p\phi_{\e^{tX}}\left( \e^{sY}\right)\right) = \zuz{\td{}{s}}{s=0}\zuz{\td{}{t}}{t=0}\underbrace{f\left( p\e^{tX}\e^{sY}\e^{-tX}\right)}_{:= F(t,s,-t)} = \\</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> = \zuz{\td{}{s}}{s=0}\zuz{\td{}{t}}{t=0}f\left( p\e^{s\phi_{\e^{tX}*}(Y)}\right) = \zuz{\td{}{s}}{s=0}\zuz{\td{}{t}}{t=0}f\left( p\phi_{\e^{tX}}\left( \e^{sY}\right)\right) = \zuz{\td{}{s}}{s=0}\zuz{\td{}{t}}{t=0}\underbrace{f\left( p\e^{tX}\e^{sY}\e^{-tX}\right)}_{:= F(t,s,-t)} = \\</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> = \zuz{\td{}{s}}{s=0}\zuz{\td{}{t}}{t=0}F(t,s,0) - \zuz{\td{}{s}}{s=0}\zuz{\td{}{t}}{t=0}F(0,s,t) = \zuz{\td{}{s}}{s=0}\zuz{\td{}{t}}{t=0}\left( f\left( p\e^{tX}\e^{sY}\right) - f\left( p\e^{sY}\e^{tX}\right)\right) = \\</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> = \zuz{\td{}{s}}{s=0}\zuz{\td{}{t}}{t=0}F(t,s,0) - \zuz{\td{}{s}}{s=0}\zuz{\td{}{t}}{t=0}F(0,s,t) = \zuz{\td{}{s}}{s=0}\zuz{\td{}{t}}{t=0}\left( f\left( p\e^{tX}\e^{sY}\right) - f\left( p\e^{sY}\e^{tX}\right)\right) = \\</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L313" >Řádka 313:</td> <td colspan="2" class="diff-lineno">Řádka 313:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\subsection{Nilpotentní a řešitelné algebry}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\subsection{Nilpotentní a řešitelné algebry}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Vet{</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Vet{</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> $\phi: \g \to \<del class="diffchange diffchange-inline">h</del>$ homomorfismus, $\h \subset \g$ podalgebra$\rimpl \left(\phi (\h) \right)^{(k)} = \phi(\h^{(k)}), \\ \left( \phi(\h) \right)^k = \phi \left( \h^k \right)$.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> $\phi: \g \to \<ins class="diffchange diffchange-inline">widetilde{g}</ins>$ homomorfismus, $\h \subset \g$ podalgebra$\rimpl \left(\phi (\h) \right)^{(k)} = \phi(\h^{(k)}), \\ \left( \phi(\h) \right)^k = \phi \left( \h^k \right)$.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> Indukcí:</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> Indukcí:</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \begin{gather*}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \begin{gather*}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> \left( \phi(\h) \right)^{(0)} = \phi(\h) <del class="diffchange diffchange-inline">\equiv </del>\phi \left( \h^{(0)} \right) \\</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> \left( \phi(\h) \right)^{(0)} = \phi(\h) <ins class="diffchange diffchange-inline">= </ins>\phi \left( \h^{(0)} \right) \\</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \left( \phi(\h) \right)^{(k)} = \left[ \phi(\h)^{(k-1)}, \phi(\h)^{(k-1)} \right] = \left[ \phi\left( \h^{(k-1)} \right), \phi \left( \h^{(k-1)} \right)\right] = \phi \left( \left[ \h^{(k-1)},\h^{(k-1)} \right] \right) = \phi\left( \h^{(k)} \right)</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \left( \phi(\h) \right)^{(k)} = \left[ \phi(\h)^{(k-1)}, \phi(\h)^{(k-1)} \right] = \left[ \phi\left( \h^{(k-1)} \right), \phi \left( \h^{(k-1)} \right)\right] = \phi \left( \left[ \h^{(k-1)},\h^{(k-1)} \right] \right) = \phi\left( \h^{(k)} \right)</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{gather*}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{gather*}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \begin{gather*}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \begin{gather*}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> \left( \phi(\h) \right)^1 = \phi(\h) <del class="diffchange diffchange-inline">\equiv </del>\phi \left( \h^1 \right) \\</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> \left( \phi(\h) \right)^1 = \phi(\h) <ins class="diffchange diffchange-inline">= </ins>\phi \left( \h^1 \right) \\</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \left( \phi(\h) \right)^k = \left[ \phi(\h)^{k-1}, \phi(\h)^{k-1} \right] = \left[ \phi\left( \h^{k-1} \right), \phi \left( \h^{k-1} \right)\right] = \phi \left( \left[ \h^{k-1},\h^{k-1} \right] \right) = \phi\left( \h^{k} \right)</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \left( \phi(\h) \right)^k = \left[ \phi(\h)^{k-1}, \phi(\h)^{k-1} \right] = \left[ \phi\left( \h^{k-1} \right), \phi \left( \h^{k-1} \right)\right] = \phi \left( \left[ \h^{k-1},\h^{k-1} \right] \right) = \phi\left( \h^{k} \right)</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{gather*} </div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{gather*} </div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L376" >Řádka 376:</td> <td colspan="2" class="diff-lineno">Řádka 376:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \item $A=S+N$,</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \item $A=S+N$,</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \item $S$ je diagonalizovatelný, $N$ nilpotentní,</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \item $S$ je diagonalizovatelný, $N$ nilpotentní,</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> \item $[S,N]=0<del class="diffchange diffchange-inline">$,</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> \item $[S,N]=0$.</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"> \item $S$ a $N$ jsou polynomy v $A</del>$.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{itemize} Bez důkazu.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{itemize} Bez důkazu.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">%</del>\<del class="diffchange diffchange-inline">Def</del>{ $<del class="diffchange diffchange-inline">A \in \gl (V)</del>$<del class="diffchange diffchange-inline">, </del>$<del class="diffchange diffchange-inline">\lambda \in \sigma (A)</del>$<del class="diffchange diffchange-inline">, </del>$<del class="diffchange diffchange-inline">V_\lambda =\lim_{n \to +\infty} \ker (</del>A<del class="diffchange diffchange-inline">-\lambda )^n</del>$ <del class="diffchange diffchange-inline">je \textbf{zobecněný vlastní podprostor $A$ příslušející $\lambda$}</del>.} </div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\<ins class="diffchange diffchange-inline">Dsl</ins>{</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">&#160; </del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"> </ins>$<ins class="diffchange diffchange-inline">S</ins>$ <ins class="diffchange diffchange-inline">a </ins>$<ins class="diffchange diffchange-inline">N</ins>$ <ins class="diffchange diffchange-inline">jsou polynomy v </ins>$A$.</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"> </ins>} <ins class="diffchange diffchange-inline"> </ins></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\subsection{Věty Lieova a Engelova} </div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\subsection{Věty Lieova a Engelova} </div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L444" >Řádka 444:</td> <td colspan="2" class="diff-lineno">Řádka 444:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \phi(X)\left( v + V_1 \right) = Xv + V_1,\qquad \forall X \in \g,\ \forall v \in V.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \phi(X)\left( v + V_1 \right) = Xv + V_1,\qquad \forall X \in \g,\ \forall v \in V.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{align}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{align}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> $\phi(\g)$ je tvořena nilpotentními operátory$\rimpl \exists e_2 \in V <del class="diffchange diffchange-inline">/ V_1</del>,\ e_2 \notin V_1,\ \phi(X)\left( e_2 + V_1 \right) = V_1,\ \forall X \in \g$, tj. $Xe_2 \in V_1,\ \forall X \in \g$. Položíme $V_2 = \mrm{span}\{ e_1, e_2 \}$ a postup opakujeme. Indukcí tedy získáváme kompoziční řadu $\{ 0 \} \subset V_1 \subset V_2 \subset \dots \subset V_n = V$ splňující $\dim V_i / V_{i-1} = 1,\ \g V_i \subset V_{i-1} \rimpl$v bázi tvořené elementy $e_i \in V_i$ jsou matice operátorů $X \in \g$ horní trojúhelníkové s nulovou diagonálou. </div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> $\phi(\g)$ je tvořena nilpotentními operátory$\rimpl \exists e_2 \in V,\ e_2 \notin V_1,\ \phi(X)\left( e_2 + V_1 \right) = V_1,\ \forall X \in \g$, tj. $Xe_2 \in V_1,\ \forall X \in \g$. Položíme $V_2 = \mrm{span}\{ e_1, e_2 \}$ a postup opakujeme. Indukcí tedy získáváme kompoziční řadu $\{ 0 \} \subset V_1 \subset V_2 \subset \dots \subset V_n = V$ splňující $\dim V_i / V_{i-1} = 1,\ \g V_i \subset V_{i-1} \rimpl$v bázi tvořené elementy $e_i \in V_i$ jsou matice operátorů $X \in \g$ horní trojúhelníkové s nulovou diagonálou. </div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{proof}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Vet{(Engelova)</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Vet{(Engelova)</div></td></tr> </table> Hazalmat https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02LIAG:Kapitola7&diff=6375&oldid=prev Hazalmat v 30. 7. 2016, 13:15 2016-07-30T13:15:58Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 30. 7. 2016, 13:15</td> </tr><tr><td colspan="2" class="diff-lineno" id="L62" >Řádka 62:</td> <td colspan="2" class="diff-lineno">Řádka 62:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> \begin{itemize}</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> \item[$\Rightarrow)$] $\g$ nilpotentní$\rimpl \exists k \leq \dim \g,\ \g^k = 0 \rimpl \g^{\dim \g-1} \subset \Zs(\g) = \zeta^1$. Indukcí ukážeme, že platí $ \g^{\dim \g - j} \subset \zeta^{j}$. Pro $j=1$ zřejmé, $j \to j+1$:</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> \begin{align*}</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> \left[ \g^{\dim \g - j - 1},\g \right] = \g^{\dim \g - j} \subset \zeta^j \rimpl \g^{\dim \g - j - 1} \subset \zeta^{j+1}.</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> \end{align*}</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> \item[$\Leftarrow)$] $\exists k,\ \zeta^k = \g$. Indukcí ukážeme, že platí $\zeta^{k-j+1} \supset \g^j$. Pro $j=1$ zřejmé, $j \to j+1$:</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> \begin{align*}</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> \g^j \subset \zeta^{k-j+1} \rimpl \left[ \g^j,\g \right] = \g^{j+1} \subset \zeta^{k-j+1-1} = \zeta^{k-j}</ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> \end{align*} </ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> $\Rightarrow\quad \g^k \subset \zeta^1 = \Zs(\g) \rimpl \g^{k+1} = 0$. </ins></div></td></tr> <tr><td colspan="2">&#160;</td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> \end{itemize}</ins></div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{proof}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Prl{</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Prl{</div></td></tr> </table> Hazalmat https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02LIAG:Kapitola7&diff=6313&oldid=prev Hazalmat v 19. 7. 2016, 05:51 2016-07-19T05:51:21Z <p></p> <table class='diff diff-contentalign-left'> <col class='diff-marker' /> <col class='diff-content' /> <col class='diff-marker' /> <col class='diff-content' /> <tr style='vertical-align: top;'> <td colspan='2' style="background-color: white; color:black; text-align: center;">← Starší verze</td> <td colspan='2' style="background-color: white; color:black; text-align: center;">Verze z 19. 7. 2016, 05:51</td> </tr><tr><td colspan="2" class="diff-lineno" id="L4" >Řádka 4:</td> <td colspan="2" class="diff-lineno">Řádka 4:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Předpokládáme konečnou dimenzi.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Předpokládáme konečnou dimenzi.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Def{</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Def{</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> <del class="diffchange diffchange-inline">Nechť $</del>\<del class="diffchange diffchange-inline">mathfrak</del>{<del class="diffchange diffchange-inline">a</del>}<del class="diffchange diffchange-inline">,</del>\<del class="diffchange diffchange-inline">mathfrak{b}</del>\<del class="diffchange diffchange-inline">subset\subset </del>\g$<del class="diffchange diffchange-inline">, </del>$[\<del class="diffchange diffchange-inline">mathfrak{a}</del>,\<del class="diffchange diffchange-inline">mathfrak{b}</del>]<del class="diffchange diffchange-inline">:=</del>\<del class="diffchange diffchange-inline">mathrm{span}</del>\<del class="diffchange diffchange-inline">{[x,y]|x\in \mathfrak{a}, y \in \mathfrak{b} \}</del>$.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> \<ins class="diffchange diffchange-inline">textbf</ins>{<ins class="diffchange diffchange-inline">Podalgebra</ins>} <ins class="diffchange diffchange-inline">$</ins>\<ins class="diffchange diffchange-inline">h$ Lieovy algebry $</ins>\<ins class="diffchange diffchange-inline">g$ je vektorový podprostor v $</ins>\g$ <ins class="diffchange diffchange-inline">splňující </ins>$[ \<ins class="diffchange diffchange-inline">h</ins>, \<ins class="diffchange diffchange-inline">h </ins>] \<ins class="diffchange diffchange-inline">subset </ins>\<ins class="diffchange diffchange-inline">h</ins>$.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\<del class="diffchange diffchange-inline">Def</del>{</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\<ins class="diffchange diffchange-inline">Pzn</ins>{</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> \<del class="diffchange diffchange-inline">textbf</del>{<del class="diffchange diffchange-inline">Podalgebra</del>} <del class="diffchange diffchange-inline">$</del>\<del class="diffchange diffchange-inline">h$ Lieovy grupy $</del>\<del class="diffchange diffchange-inline">g$ je vektorový podprostor v $</del>\g$ <del class="diffchange diffchange-inline">splňující </del>$[ \<del class="diffchange diffchange-inline">h</del>, \<del class="diffchange diffchange-inline">h </del>] \<del class="diffchange diffchange-inline">subset </del>\<del class="diffchange diffchange-inline">h</del>$.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> <ins class="diffchange diffchange-inline">Nechť $</ins>\<ins class="diffchange diffchange-inline">mathfrak</ins>{<ins class="diffchange diffchange-inline">a</ins>}<ins class="diffchange diffchange-inline">,</ins>\<ins class="diffchange diffchange-inline">mathfrak{b}</ins>\<ins class="diffchange diffchange-inline">subset\subset </ins>\g$<ins class="diffchange diffchange-inline">, </ins>$[\<ins class="diffchange diffchange-inline">mathfrak{a}</ins>,\<ins class="diffchange diffchange-inline">mathfrak{b}</ins>]<ins class="diffchange diffchange-inline">=</ins>\<ins class="diffchange diffchange-inline">mathrm{span}</ins>\<ins class="diffchange diffchange-inline">{[X,Y]|X\in \mathfrak{a}, Y \in \mathfrak{b} \}</ins>$.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Prl{</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Prl{</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L43" >Řádka 43:</td> <td colspan="2" class="diff-lineno">Řádka 43:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>&#160; &#160;</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>&#160; &#160;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\subsection<del class="diffchange diffchange-inline">*</del>{Charakteristické série ideálů} </div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\subsection{Charakteristické série ideálů} </div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Def{</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Def{</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \textbf{Centrum} Lieovy algebry $\g$&#160; je maximální ideál $\zeta^1$ s~vlastností $[\g ,\zeta^1]=0$, tj. $\zeta^1=\{x\in \g |[x,\g]=0 \}$, značíme $\Zs (\g)=\zeta(\g)=\zeta^1 (\g) =\zeta^1$.</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \textbf{Centrum} Lieovy algebry $\g$&#160; je maximální ideál $\zeta^1$ s~vlastností $[\g ,\zeta^1]=0$, tj. $\zeta^1=\{x\in \g |[x,\g]=0 \}$, značíme $\Zs (\g)=\zeta(\g)=\zeta^1 (\g) =\zeta^1$.</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L56" >Řádka 56:</td> <td colspan="2" class="diff-lineno">Řádka 56:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> } </div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> } </div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Pzn{</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Pzn{</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> $\g^2=\g^{(1)}$, $\g^{(k)}\subset \g^k+1$, tj. každá nilpotentní $\g$ je řešitelná.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> $\g^2=\g^{(1)}$, $\g^{(k)}\subset \g^<ins class="diffchange diffchange-inline">{</ins>k+1<ins class="diffchange diffchange-inline">}</ins>$, tj. každá nilpotentní $\g$ je řešitelná.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Vet{</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\Vet{</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L98" >Řádka 98:</td> <td colspan="2" class="diff-lineno">Řádka 98:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \g=\s \dotplus \rr ,\ [\s,\s]\subset \s ,\ [\s,\rr]\subset \rr ,\ [\rr,\rr]\subset \rr ,</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \g=\s \dotplus \rr ,\ [\s,\s]\subset \s ,\ [\s,\rr]\subset \rr ,\ [\rr,\rr]\subset \rr ,</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{align*}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{align*}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> přičemž $\s$ <del class="diffchange diffchange-inline">ke </del>určena až na izomorfii. $\s$ nebo $\rr$ může být rovna $0$. Bez důkazu.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> přičemž $\s$ <ins class="diffchange diffchange-inline">je </ins>určena až na izomorfii. $\s$ nebo $\rr$ může být rovna $0$. Bez důkazu.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\subsection{Vlastnosti ideálů (cvičení)}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\subsection{Vlastnosti ideálů (cvičení)}</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L128" >Řádka 128:</td> <td colspan="2" class="diff-lineno">Řádka 128:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> } </div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> } </div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> Chceme ukázat, že $\exists n \in \N,\ \forall <del class="diffchange diffchange-inline">x_1</del>,\dots,<del class="diffchange diffchange-inline">x_n </del>\in \h_1 \cup \h_2,\ [<del class="diffchange diffchange-inline">x_1</del>,[<del class="diffchange diffchange-inline">x_2</del>,\dots,[<del class="diffchange diffchange-inline">x_</del>{n-1},<del class="diffchange diffchange-inline">x_n</del>]]] = 0$.\\</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> Chceme ukázat, že $\exists n \in \N,\ \forall <ins class="diffchange diffchange-inline">X_1</ins>,\dots,<ins class="diffchange diffchange-inline">X_n </ins>\in \h_1 \cup \h_2,\ [<ins class="diffchange diffchange-inline">X_1</ins>,[<ins class="diffchange diffchange-inline">X_2</ins>,\dots,[<ins class="diffchange diffchange-inline">X_</ins>{n-1},<ins class="diffchange diffchange-inline">X_n</ins>]]] = 0$.\\</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> $\h_1,\h_2$ nilpotentní $\Leftrightarrow \exists k \in \N,\ \h_1^k =0,\ \h_2^k = 0$, vezmeme tedy $n = 2k \rimpl$BÚNO aspoň $k$ z~$<del class="diffchange diffchange-inline">x_1</del>,\dots,<del class="diffchange diffchange-inline">x_n</del>$ je z $\h_1$:</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> $\h_1,\h_2$ nilpotentní $\Leftrightarrow \exists k \in \N,\ \h_1^k =0,\ \h_2^k = 0$, vezmeme tedy $n = 2k \rimpl$BÚNO aspoň $k$ z~$<ins class="diffchange diffchange-inline">X_1</ins>,\dots,<ins class="diffchange diffchange-inline">X_n</ins>$ je z $\h_1$:</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \begin{align*}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \begin{align*}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> <del class="diffchange diffchange-inline">x </del>\in \h_1^j,\ <del class="diffchange diffchange-inline">y </del>\in \h_1 \rimpl [<del class="diffchange diffchange-inline">x</del>,<del class="diffchange diffchange-inline">y</del>] \in \h_1^{j+1},&amp;&amp; <del class="diffchange diffchange-inline">x </del>\in \h_1^j,\ <del class="diffchange diffchange-inline">y </del>\in \h_2 \rimpl [<del class="diffchange diffchange-inline">x</del>,<del class="diffchange diffchange-inline">y</del>] \in \h_1^j</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> <ins class="diffchange diffchange-inline">X </ins>\in \h_1^j,\ <ins class="diffchange diffchange-inline">Y </ins>\in \h_1 \rimpl [<ins class="diffchange diffchange-inline">X</ins>,<ins class="diffchange diffchange-inline">Y</ins>] \in \h_1^{j+1},&amp;&amp; <ins class="diffchange diffchange-inline">X </ins>\in \h_1^j,\ <ins class="diffchange diffchange-inline">Y </ins>\in \h_2 \rimpl [<ins class="diffchange diffchange-inline">X</ins>,<ins class="diffchange diffchange-inline">Y</ins>] \in \h_1^j</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{align*} &#160;</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{align*} &#160;</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> $\Rightarrow \quad [<del class="diffchange diffchange-inline">x_1</del>,[<del class="diffchange diffchange-inline">x_2</del>,\dots,[<del class="diffchange diffchange-inline">x_</del>{n-1},<del class="diffchange diffchange-inline">x_n</del>]]] \in \h_1^k = \{ 0\}$ </div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> $\Rightarrow \quad [<ins class="diffchange diffchange-inline">X_1</ins>,[<ins class="diffchange diffchange-inline">X_2</ins>,\dots,[<ins class="diffchange diffchange-inline">X_</ins>{n-1},<ins class="diffchange diffchange-inline">X_n</ins>]]] \in \h_1^k = \{ 0\}$ </div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{proof} </div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{proof} </div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>&#160; &#160;</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>&#160; &#160;</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L426" >Řádka 426:</td> <td colspan="2" class="diff-lineno">Řádka 426:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{proof}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\lmma{</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\lmma{</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> Buď $\g \subset \gl(V),\ \forall X \in \g$ nilpotenti, tj Lieova algebra nilpotentních operátorů. Pak $\exists \varepsilon =&#160; \{ e_i \}$ báze taková že $Xe_i \in \mrm{span}\{ e_1,\dots,e_{i-1} \},\ \forall X \in \g$, tj. $\forall X \in \g,\ X_\varepsilon$ horní trojúhelníková matice, tj. $\g$ je nilpotentní algebra. &#160;</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> Buď $\g \subset \gl(V),\ \forall X \in \g$ nilpotenti, tj Lieova algebra nilpotentních operátorů. Pak $\exists \varepsilon =&#160; \{ e_i \}$ báze taková že $Xe_i \in \mrm{span}\{ e_1,\dots,e_{i-1} \},\ \forall X \in \g$, tj. $\forall X \in \g,\ X_\varepsilon$ <ins class="diffchange diffchange-inline">ostře </ins>horní trojúhelníková matice, tj. $\g$ je nilpotentní algebra. &#160;</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> } </div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> } </div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="L491" >Řádka 491:</td> <td colspan="2" class="diff-lineno">Řádka 491:</td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \left( X - \lambda \mathbb{1} \right) ^k Y W_{\lambda}^X= \sum_{j=0}^k\binom{k}{j}\left( \ad_X \right)^j Y \left( X - \lambda \mathbb{1} \right)^{k-j}W_\lambda^X \quad \xrightarrow{k \to +\infty} \quad 0,\qquad \forall Y \in \g. </div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \left( X - \lambda \mathbb{1} \right) ^k Y W_{\lambda}^X= \sum_{j=0}^k\binom{k}{j}\left( \ad_X \right)^j Y \left( X - \lambda \mathbb{1} \right)^{k-j}W_\lambda^X \quad \xrightarrow{k \to +\infty} \quad 0,\qquad \forall Y \in \g. </div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{align*}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{align*}</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> Protože pro dostatečně velké $k$ je buď $\ad_X^jY = 0$ nebo $(X - \lambda\mathbb{1})^{k-j}W_\lambda^X = 0 \rimpl YW_\lambda^X \subset W_\lambda^X$, tj. $W_\lambda^X$ je invariantní podprostor. </div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> Protože pro dostatečně velké $k$ je buď $<ins class="diffchange diffchange-inline">(</ins>\ad_X<ins class="diffchange diffchange-inline">)</ins>^jY = 0$ nebo $(X - \lambda\mathbb{1})^{k-j}W_\lambda^X = 0 \rimpl YW_\lambda^X \subset W_\lambda^X$, tj. $W_\lambda^X$ je invariantní podprostor. </div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> \end{proof}</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\lmma{</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\lmma{</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> Buď $V$ nad $\C, \g \subset \gl(V)$ řešitelná. Pak $\exists v \in V, \ v \neq 0,\ \lambda \in g^*,\ \forall X \in \g,\ Xv = \lambda(X)v$.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> Buď $V$ nad $\C,<ins class="diffchange diffchange-inline">\ </ins>\g \subset \gl(V)$ řešitelná. Pak $\exists v \in V, \ v \neq 0,\ \lambda \in g^*,\ \forall X \in \g,\ Xv = \lambda(X)v$.</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div> }</div></td></tr> <tr><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td><td class='diff-marker'>&#160;</td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{proof}</div></td></tr> </table> Hazalmat https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02LIAG:Kapitola7&diff=6299&oldid=prev Hazalmat v 16. 7. 2016, 14:34 2016-07-16T14:34:15Z <p></p> <a href="https://wikiskripta.fjfi.cvut.cz/wiki/index.php?title=02LIAG:Kapitola7&amp;diff=6299&amp;oldid=6247">Ukázat změny</a> Hazalmat