<s>[INST] Linear Transformation textbook problem A problem from my Linear Algebra textbook:
Let $T$ be a linear transformation from $M_{2,2}$ into $M_{2,2}$ such that
$T\begin{pmatrix}
\begin{bmatrix} 1&0 \\ 0&0 \end{bmatrix}\end{pmatrix}
=
\begin{bmatrix} 1&-1 \\ 0&2 \end{bmatrix}$, 
$T\begin{pmatrix}
\begin{bmatrix} 0&1 \\ 0&0 \end{bmatrix}\end{pmatrix}
=
\begin{bmatrix} 0&2 \\ 1&1 \end{bmatrix}$, 
$T\begin{pmatrix}
\begin{bmatrix} 0&0 \\ 1&0 \end{bmatrix}\end{pmatrix}
=
\begin{bmatrix} 1&2 \\ 0&1 \end{bmatrix}$, 
$T\begin{pmatrix}
\begin{bmatrix} 0&0 \\ 0&1 \end{bmatrix}\end{pmatrix}
=
\begin{bmatrix} 3&-1 \\ 1&0 \end{bmatrix}$.
Find $T\begin{pmatrix}
\begin{bmatrix} 1&3 \\ -1&4 \end{bmatrix}\end{pmatrix}$
I just need to know how to start this.  I tried setting it up like this:
$\begin{bmatrix} a&b \\ c&d \end{bmatrix}$
$\begin{bmatrix} 1&0 \\ 0&0 \end{bmatrix}$
= $\begin{bmatrix} 1&-1 \\ 0&2 \end{bmatrix}$
which gets me to here:
$\begin{bmatrix} a&0 \\ c&0 \end{bmatrix}$
= $\begin{bmatrix} 1&-1 \\ 0&2 \end{bmatrix}$
and then I'm lost...
 [/INST] Use linearity:
$$T\left(\sum_{i=1}^n \alpha_i A_i\right) = \sum_{i=1}^n \alpha_i T(A_i)$$
Also, note that a linear transformation from $M_{2,2}$ to $M_{2,2}$ is not a $2\times 2$ matrix itself.
Finally, note that
$$\begin{align*}
\pmatrix{1&3\\-1&4} & = \pmatrix{1&0\\0&0} + 3 \pmatrix{0&1\\0&0} - \pmatrix{0&0\\1&0} + 4 \pmatrix{0&0\\0&1} \\
\Rightarrow T\pmatrix{1&3\\-1&4} & = T\pmatrix{1&0\\0&0} + 3T\pmatrix{0&1\\0&0} - T\pmatrix{0&0\\1&0} +4T\pmatrix{0&0\\0&1}\\
& = \pmatrix{1&-1\\0&2} + 3\pmatrix{0&2\\1&1} - \pmatrix{1&2\\0&1} + 4\pmatrix{3&-1\\1&0} \\
& = \pmatrix{12&-1\\7&4}
\end{align*}$$
</s>