<s>[INST] Linear transformation matrix of vector space $\mathbb R^{2x2}$ 
Select a basis B of a vector space $\mathbb R^{2x2}$ and for linear transformation $f:\mathbb R^{2x2}→\mathbb R^{2x2}$ given by $f(X) = \begin{pmatrix}1 & 0 \\ 1 & 0 \end{pmatrix}X+\begin{pmatrix}1 & 1 \\ 1 & 1 \end{pmatrix}X^T$ compute the matrix relative to the base B.


I selected canonical basis $B=\{\begin{pmatrix}1 & 0 \\ 0 & 0 \end{pmatrix},\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix},\begin{pmatrix}0 & 0 \\ 1 & 0 \end{pmatrix},\begin{pmatrix}0 & 0 \\ 0 & 1 \end{pmatrix}\}$
Then I computed transformation of each vector in the basis according to the task.
Got $f(B)=\{\begin{pmatrix}2 & 0 \\ 2 & 0 \end{pmatrix},\begin{pmatrix}1 & 1 \\ 1 & 1 \end{pmatrix},\begin{pmatrix}0 & 1 \\ 0 & 1 \end{pmatrix},\begin{pmatrix}0 & 1 \\ 0 & 1 \end{pmatrix}\}$, if I'm not wrong.
What should I do now?
 [/INST] If it's indeed the case that
$$f\begin{pmatrix}1 & 0 \\ 0 & 0 \end{pmatrix} = 2 \begin{pmatrix}1 & 0 \\ 0 & 0 \end{pmatrix} + 2 \begin{pmatrix}0 & 0 \\ 1 & 0 \end{pmatrix}$$
(I didn't check your calculations), then this would tell us that the first row of the matrix is $2, 0, 2, 0$, and so forth.
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