master > master: Formattierung
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@ -5202,9 +5202,9 @@ Seien $n\in\ntrlpos$ und $\mathbf{v}_{i}\in V$ für $i\in\{1,2,\ldots,n\}$.
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&x_{1}-x_{2}-2x_{3}+2x_{4}=0
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&\Longleftrightarrow
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&\underbrace{
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\begin{smatrix}
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1&-1&-2&2\\
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\end{smatrix}
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\begin{matrix}{cccc}
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1 &-1 &-2 &2\\
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\end{matrix}
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}_{=:A_{1}}
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\mathbf{x}=\zerovector\\
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\mathbf{x}\in U_{2}
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@ -5212,9 +5212,9 @@ Seien $n\in\ntrlpos$ und $\mathbf{v}_{i}\in V$ für $i\in\{1,2,\ldots,n\}$.
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&x_{1}-x_{2}-x_{3}-x_{4}=0
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&\Longleftrightarrow
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&\underbrace{
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\begin{smatrix}
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1&-1&-1&-1\\
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\end{smatrix}
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\begin{matrix}{cccc}
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1 &-1 &-1 &-1\\
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\end{matrix}
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}_{=:A_{2}}
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\mathbf{x}=\zerovector\\
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\mathbf{x}\in U_{1}\cap U_{2}
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@ -5225,10 +5225,10 @@ Seien $n\in\ntrlpos$ und $\mathbf{v}_{i}\in V$ für $i\in\{1,2,\ldots,n\}$.
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\end{array}
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&\Longleftrightarrow
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&\underbrace{
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\begin{smatrix}
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1&-1&-2&2\\
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1&-1&-1&-1\\
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\end{smatrix}
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\begin{matrix}{cccc}
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1 &-1 &-2 &2\\
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1 &-1 &-1 &-1\\
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\end{matrix}
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}_{=:A_{3}}
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\mathbf{x}=\zerovector\\
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\end{longmathe}
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@ -5245,9 +5245,9 @@ Seien $n\in\ntrlpos$ und $\mathbf{v}_{i}\in V$ für $i\in\{1,2,\ldots,n\}$.
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Zeilenstufenform für $A_{1}$:
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\begin{mathe}[mc]{rcl}
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A_{1} &= &\begin{smatrix}
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1&-1&-2&2\\
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\end{smatrix}\\
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A_{1} &= &\begin{matrix}{cccc}
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1 &-1 &-2 &2\\
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\end{matrix}\\
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\end{mathe}
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Darum sind $x_{2}$, $x_{3}$, $x_{4}$ frei und $x_{1}$ wird durch diese bestimmt.
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@ -5281,12 +5281,12 @@ Seien $n\in\ntrlpos$ und $\mathbf{v}_{i}\in V$ für $i\in\{1,2,\ldots,n\}$.
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U_{1}
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&= &\{\mathbf{x}\in V\mid A_{1}\mathbf{x}=\zerovector\}
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&= &\vectorspacespan\underbrace{
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\{
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\left\{
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\begin{svector}1\\1\\0\\0\\\end{svector},
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\begin{svector}2\\0\\1\\0\\\end{svector},
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\begin{svector}-2\\0\\0\\1\\\end{svector}
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\}
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}_{=:B}\\
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\right\}
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}_{=:B_{1}}\\
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\end{mathe}
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und \fbox{$B_{1}$ bildet eine Basis für $U_{1}$}.
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@ -5295,9 +5295,9 @@ Seien $n\in\ntrlpos$ und $\mathbf{v}_{i}\in V$ für $i\in\{1,2,\ldots,n\}$.
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Zeilenstufenform für $A_{2}$:
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\begin{mathe}[mc]{rcl}
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A_{2} &= &\begin{smatrix}
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1&-1&-2&2\\
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\end{smatrix}\\
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A_{2} &= &\begin{matrix}{cccc}
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1 &-1 &-2 &2\\
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\end{matrix}\\
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\end{mathe}
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Darum sind $x_{2}$, $x_{3}$, $x_{4}$ frei und $x_{1}$ wird durch diese bestimmt.
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@ -5329,11 +5329,11 @@ Seien $n\in\ntrlpos$ und $\mathbf{v}_{i}\in V$ für $i\in\{1,2,\ldots,n\}$.
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U_{2}
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&= &\{\mathbf{x}\in V\mid A_{2}\mathbf{x}=\zerovector\}
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&= &\vectorspacespan\underbrace{
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\{
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\left\{
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\begin{svector}1\\1\\0\\0\\\end{svector},
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\begin{svector}1\\0\\1\\0\\\end{svector},
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\begin{svector}1\\0\\0\\1\\\end{svector}
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\}
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\right\}
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}_{=:B_{2}}\\
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\end{mathe}
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@ -5344,14 +5344,14 @@ Seien $n\in\ntrlpos$ und $\mathbf{v}_{i}\in V$ für $i\in\{1,2,\ldots,n\}$.
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\begin{mathe}[mc]{rclcl}
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A_{3}
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&= &\begin{smatrix}
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1&-1&-2&2\\
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1&-1&-1&-1\\
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\end{smatrix}
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&\rightsquigarrow &\begin{smatrix}
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1&-1&-2&2\\
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0&0&1&-3\\
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\end{smatrix}\\
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&= &\begin{matrix}{cccc}
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1 &-1 &-2 &2\\
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1 &-1 &-1 &-1\\
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\end{matrix}
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&\rightsquigarrow &\begin{matrix}{cccc}
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1 &-1 &-2 &2\\
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0 &0 &1 &-3\\
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\end{matrix}\\
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\end{mathe}
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Darum sind $x_{2}$, $x_{4}$, frei und $x_{1}$, $x_{3}$ werden durch diese bestimmt.
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@ -5385,10 +5385,10 @@ Seien $n\in\ntrlpos$ und $\mathbf{v}_{i}\in V$ für $i\in\{1,2,\ldots,n\}$.
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U_{1}\cap U_{2}
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&= &\{\mathbf{x}\in V\mid A_{3}\mathbf{x}=\zerovector\}
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&= &\vectorspacespan\underbrace{
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\{
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\left\{
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\begin{svector}1\\1\\0\\0\\\end{svector},
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\begin{svector}4\\0\\3\\1\\\end{svector}
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\}
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\right\}
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}_{=:B_{3}}\\
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\end{mathe}
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@ -5401,23 +5401,23 @@ Seien $n\in\ntrlpos$ und $\mathbf{v}_{i}\in V$ für $i\in\{1,2,\ldots,n\}$.
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\begin{mathe}[mc]{rcl}
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U_{1}+U_{2}
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&= &\vectorspacespan\{
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&= &\vectorspacespan\big\{
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\begin{svector}1\\1\\0\\0\\\end{svector},
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\begin{svector}2\\0\\1\\0\\\end{svector},
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\begin{svector}-2\\0\\0\\1\\\end{svector}
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\}
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+\vectorspacespan\{
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\big\}
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+\vectorspacespan\big\{
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\begin{svector}1\\1\\0\\0\\\end{svector},
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\begin{svector}1\\0\\1\\0\\\end{svector},
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\begin{svector}1\\0\\0\\1\\\end{svector}
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\}\\
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&= &\vectorspacespan\{
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\big\}\\
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&= &\vectorspacespan\big\{
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\begin{svector}1\\1\\0\\0\\\end{svector},
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\begin{svector}2\\0\\1\\0\\\end{svector},
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\begin{svector}-2\\0\\0\\1\\\end{svector}
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\begin{svector}1\\0\\1\\0\\\end{svector},
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\begin{svector}1\\0\\0\\1\\\end{svector}
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\}.\\
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\big\}.\\
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\end{mathe}
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Wir haben nun ein Erzeugendensystem bestimmt.
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@ -5433,12 +5433,12 @@ Seien $n\in\ntrlpos$ und $\mathbf{v}_{i}\in V$ für $i\in\{1,2,\ldots,n\}$.
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Homogenes System:
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\begin{mathe}[mc]{rcl}
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\begin{smatrix}
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1&2&-2&1&1\\
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1&0&0&0&0\\
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0&1&0&1&0\\
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0&0&1&0&1\\
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\end{smatrix}\\
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\begin{matrix}{ccccc}
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1 &2 &-2 &1 &1\\
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1 &0 &0 &0 &0\\
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0 &1 &0 &1 &0\\
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0 &0 &1 &0 &1\\
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\end{matrix}\\
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\end{mathe}
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Zeilenoperation
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@ -5446,12 +5446,12 @@ Seien $n\in\ntrlpos$ und $\mathbf{v}_{i}\in V$ für $i\in\{1,2,\ldots,n\}$.
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anwenden:
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\begin{mathe}[mc]{rcl}
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\begin{smatrix}
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1&2&-2&1&1\\
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0&2&-2&1&1\\
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0&1&0&1&0\\
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0&0&1&0&1\\
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\end{smatrix}\\
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\begin{matrix}{ccccc}
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1 &2 &-2 &1 &1\\
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0 &2 &-2 &1 &1\\
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0 &1 &0 &1 &0\\
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0 &0 &1 &0 &1\\
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\end{matrix}\\
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\end{mathe}
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Zeilenoperation
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@ -5459,12 +5459,12 @@ Seien $n\in\ntrlpos$ und $\mathbf{v}_{i}\in V$ für $i\in\{1,2,\ldots,n\}$.
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anwenden:
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\begin{mathe}[mc]{rcl}
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\begin{smatrix}
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1&2&-2&1&1\\
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0&2&-2&1&1\\
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0&0&2&1&-1\\
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0&0&1&0&1\\
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\end{smatrix}\\
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\begin{matrix}{ccccc}
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1 &2 &-2 &1 &1\\
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0 &2 &-2 &1 &1\\
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0 &0 &2 &1 &-1\\
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0 &0 &1 &0 &1\\
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\end{matrix}\\
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\end{mathe}
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Zeilenoperation
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@ -5472,12 +5472,12 @@ Seien $n\in\ntrlpos$ und $\mathbf{v}_{i}\in V$ für $i\in\{1,2,\ldots,n\}$.
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anwenden:
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\begin{mathe}[mc]{rcl}
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\begin{smatrix}
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1&2&-2&1&1\\
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0&2&-2&1&1\\
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0&0&2&1&-1\\
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0&0&0&1&-3\\
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\end{smatrix}\\
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\begin{matrix}{ccccc}
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1 &2 &-2 &1 &1\\
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0 &2 &-2 &1 &1\\
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0 &0 &2 &1 &-1\\
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0 &0 &0 &1 &-3\\
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\end{matrix}\\
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\end{mathe}
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$\Longrightarrow$ nur $x_{5}$ frei.
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