How do I find the vector $T\begin{pmatrix} 5 & 0 \\ -10 & -13 \end{pmatrix}$? I defined a function $T: M^R_{2x2} \rightarrow R_4[x]$ and I defined:
$T\begin{pmatrix} 2 & 3 \\ 1 & 0 \end{pmatrix} = x^2$
$T\begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} = 3x - 4$
$T\begin{pmatrix} 0 & 2 \\ 4 & 5 \end{pmatrix} = 2x^2 - 7$
How can I find the following function if I know that all three vectors above are linearly independent:
$T\begin{pmatrix} 5 & 0 \\ -10 & -13 \end{pmatrix} = ?$
What is the method to handle such questions?

Hint: You'll need to determine whether there exist $a,b,c$ for which
$$
\pmatrix{5&0\\-10&-13} = 
a\pmatrix{2&3\\1&0} + 
b\pmatrix{1&0\\0&2} + 
c \pmatrix{0&2\\4&5}
$$
and, if such $a,b,c$ exist, find them.
