# How to write a linear map as a matrix with respect to a given canonical basis

I am asked to write a linear map as a matrix with respect to a given canonical basis. The basis is

$b = \left \{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right \}$.

The map is given by $\phi: \mathbb{R}^2 \rightarrow \mathbb{R^2}$; $(x,y) \rightarrow (x+y, x-y)$.

I know that $\phi$ as a matrix is $\begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$.

Any help would be nice. Thank you for your time.

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Study the answer here: math.stackexchange.com/a/608674/52893 –  JohnD Jan 21 '14 at 16:39

The column vectors of the matrix of $\phi$ are the images of the basis vectors under $\phi$. For example the image of the first basis vector is $\phi\begin{pmatrix}1\\ 0\end{pmatrix}=\begin{pmatrix}1+0\\ 1-0\end{pmatrix}=\begin{pmatrix}1\\ 1\end{pmatrix}$, which is the first column of the matrix of $\phi$.