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linear_algebra:04_invertible_matrices [2014/01/13 20:52]
marje created
linear_algebra:04_invertible_matrices [2014/01/20 13:39] (current)
marje
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 </​box>​ </​box>​
  
-<​WRAP ​indent+<​WRAP ​task
-:?: Verify the lemma in case of $A=\begin{pmatrix}1&​3&​2&​4\\-4&​-2&​-3&​-1\\5&​8&​6&​7\end{pmatrix}$ by calculating+Verify the lemma in case of $A=\begin{pmatrix}1&​3&​2&​4\\-4&​-2&​-3&​-1\\5&​8&​6&​7\end{pmatrix}$ by calculating
  
 ;#; ;#;
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 means just calculating the product $Db$.  means just calculating the product $Db$. 
  
-<​WRAP ​indent+<​WRAP>​ 
-:?: Let $A=\begin{pmatrix}1&​0&​1\\1&​-1&​1\\1&​1&​-1\end{pmatrix},​\ \mathbf{b}=\begin{pmatrix}1\\2\\3\end{pmatrix}$,​+Let $A=\begin{pmatrix}1&​0&​1\\1&​-1&​1\\1&​1&​-1\end{pmatrix},​\ \mathbf{b}=\begin{pmatrix}1\\2\\3\end{pmatrix}$,​
 and $D=\begin{pmatrix}0&​\frac{1}{2}&​\frac{1}{2}\\1&​-1&​0\\1&​-\frac{1}{2}&​-\frac{1}{2}\end{pmatrix}$.\\ and $D=\begin{pmatrix}0&​\frac{1}{2}&​\frac{1}{2}\\1&​-1&​0\\1&​-\frac{1}{2}&​-\frac{1}{2}\end{pmatrix}$.\\
 Verify that $DA=I$ and solve the system $A\mathbf{x}=\mathbf{b}$ by usind the matrix $D$.\\ Verify that $DA=I$ and solve the system $A\mathbf{x}=\mathbf{b}$ by usind the matrix $D$.\\
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 we can solve the system easily for many different values of $\mathbf{b}$. we can solve the system easily for many different values of $\mathbf{b}$.
  
-<​WRAP ​indent> +<​WRAP ​task>
-:?:+
 Solve the system of linear equations form previous exercise for  Solve the system of linear equations form previous exercise for 
 $\mathbf{b}=\begin{pmatrix}0\\1\\3\end{pmatrix}$ and $\mathbf{b}=\begin{pmatrix}1\\1\\3\end{pmatrix}$. $\mathbf{b}=\begin{pmatrix}0\\1\\3\end{pmatrix}$ and $\mathbf{b}=\begin{pmatrix}1\\1\\3\end{pmatrix}$.
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 Thus, for solving the system $A\mathbf{x}=\mathbf{b}$,​ we are looking for a $n\times n$ matrix $D$  Thus, for solving the system $A\mathbf{x}=\mathbf{b}$,​ we are looking for a $n\times n$ matrix $D$ 
 such that $AD=I$. It is a matrix equation where $D$ is the unknown. From previous we know  such that $AD=I$. It is a matrix equation where $D$ is the unknown. From previous we know 
-(see [[linear_algebra:​gaussian_elimination|Gaussian elimination algorithm]]) how to solved ​+(see [[linear_algebra:​03_gaussian_elimination|Gaussian elimination algorithm]]) how to solved ​
 similar equations where the unknown is and $n\times 1$ vector and on the right hand side is also an vector ​ similar equations where the unknown is and $n\times 1$ vector and on the right hand side is also an vector ​
 - this is were we are heading. - this is were we are heading.
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 \qquad (\ast)$$ \qquad (\ast)$$
  
-<​WRAP ​indent> +<​WRAP ​task>
-:?:+
 Calculate the second column of the product Calculate the second column of the product
  
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 $$ $$
  
-<​WRAP ​indent> +<​WRAP ​task>
-:?:+
 Perform the above mentioned elementary row operations on the matrix Perform the above mentioned elementary row operations on the matrix
  
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 system even tough $A^{-1}$ is not calculated. system even tough $A^{-1}$ is not calculated.
  
-<​WRAP ​indent> +<​WRAP ​task>
-:?:+
 Find the inverse of the matrix Find the inverse of the matrix
 \begin{align*} \begin{align*}
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 </​WRAP>​ </​WRAP>​
  
-<​WRAP ​indent> +<​WRAP ​task>
-:?:+
 Modify your IPython script for simple Gaussian elimination algorithm for finding the inverse of a given matrix. ​ Modify your IPython script for simple Gaussian elimination algorithm for finding the inverse of a given matrix. ​
 </​WRAP>​ </​WRAP>​
linear_algebra/04_invertible_matrices.txt · Last modified: 2014/01/20 13:39 by marje