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  • Let {$e_1,e_2$} be a standard basis of $\mathbb{R^2}$. Suppose $L:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ be a linear transformation such that $L\begin{pmatrix} 1 \\ 1 \end{pmatrix}=\begin{pmatrix} 3 \\ 3 \end{pmatrix}$ and $L\begin{pmatrix} 1 \\ -1 \end{pmatrix}=\begin{pmatrix} -1 \\ 1 \end{pmatrix}$. Find the matrix representation of $L$ with respect to the standard basis.

I couldn't anything. Can you help, can you give a hint?

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    $\begingroup$ Represent the standard basis vectors as linear combinations of $(1,1)$ and $(1,-1)$. $\endgroup$ – Friedrich Philipp Nov 11 '17 at 20:51
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Here's one approach. Let $\begin{pmatrix}a & b \\ c & d\end{pmatrix}$ be the matrix of $L$ with respect to the standard basis. Then $L\begin{pmatrix} 1 \\ 1 \end{pmatrix}=\begin{pmatrix} 3 \\ 3 \end{pmatrix}$ and $L\begin{pmatrix} 1 \\ -1 \end{pmatrix}=\begin{pmatrix} -1 \\ 1 \end{pmatrix}$ give the equations $$a+b=3, \quad c+d=3, \quad a-b=-1, \quad c-d=1\,.$$ Solve for $a,b,c,d$.

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  • $\begingroup$ Why $a-b=-1$ and $c-d=1$? $\endgroup$ – PozcuKushimotoStreet Nov 11 '17 at 20:57
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    $\begingroup$ $\begin{pmatrix} a & b \\ c&d \end{pmatrix}\begin{pmatrix} 1 \\ -1 \end{pmatrix}=\begin{pmatrix} -1 \\ 1 \end{pmatrix}$ becomes $\begin{pmatrix} a - b \\ c - d\end{pmatrix}=\begin{pmatrix} -1 \\ 1 \end{pmatrix}$ $\endgroup$ – A. Goodier Nov 11 '17 at 21:00
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    $\begingroup$ Replace $L$ with the matrix $\begin{pmatrix}a & b \\ c & d\end{pmatrix}$ in $L\begin{pmatrix} 1 \\ 1 \end{pmatrix}=\begin{pmatrix} 3 \\ 3 \end{pmatrix}$ and $L\begin{pmatrix} 1 \\ -1 \end{pmatrix}=\begin{pmatrix} -1 \\ 1 \end{pmatrix}$ and do the matrix multiplication on the left hand side $\endgroup$ – A. Goodier Nov 11 '17 at 21:01
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    $\begingroup$ Yes, that's right $\endgroup$ – A. Goodier Nov 11 '17 at 21:05
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    $\begingroup$ The matrix of a linear transformation $L:\mathbb{R}^2\to\mathbb{R}^2$ is the $2\times 2$ matrix $M$, such that $M\begin{pmatrix}x \\ y\end{pmatrix}=L(x,y)$ for every $(x,y)\in\mathbb{R}^2$. We know $L(x,y)$ for some specific points $(x,y)\in\mathbb{R}^2$. These enable us to determine the matrix. $\endgroup$ – A. Goodier Nov 11 '17 at 21:32

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