<s>[INST] Find Matrix T of a Linear Transformation $T: M_{2\times 2} \to P_3$ Been staring at this practice problem and scribbling nonsense for quite a while now googled extensively and such but nothing triggered any spark of understanding.
All in all a helpful shove off the correct cliff would be appreciated.
Consider the linear transformation $T: M_{2\times 2} \to P_3$ given by
$$
T\left ( \begin{bmatrix}
a & b \\ 
c & d
\end{bmatrix} \right ) = (a + b + c) +(a - b - c)x + (a + d)x^2 + (b + c - d)x^3
 $$
(a) Find the matix of T with respect to the usual bases for $M_{2x2}$ and $P_3$
(b) Now consider the basis $B$ of $M_{2x2}$ given by
$$
B =\left\{
\begin{bmatrix}1 & 0 \\0 & 0 \end{bmatrix},
\begin{bmatrix}0 & 1 \\1 & 0 \end{bmatrix},
\begin{bmatrix}0 & 1 \\-1 & 0 \end{bmatrix},
\begin{bmatrix}0 & 0 \\0 & 1 \end{bmatrix}
\right\}
$$
and the basis of $C$ of $P_3$ given by 
$C =\left\{ 1+x+x^2, 1-x+x^3, x^2+x^3,x^2-x^3\right\}$ 
find the matrix $\left[T\right]_{C\leftarrow B}$
 [/INST] For the $2\times 2$ matrices, the standard basis would be
$$
\begin{pmatrix} 1 & 0 \\ 0&0   \end{pmatrix}, \begin{pmatrix} 0&1 \\ 0&0 \end{pmatrix}, \begin{pmatrix} 0&0\\1&0 \end{pmatrix}, \begin{pmatrix} 0&0\\0&1 \end{pmatrix}
$$
and for the polynomials the standard basis is $\{1, x, x^2, x^3\}$. Now you want to see what $T$ does to the standard basis of $M_{22}$ and write it in terms of the standard basis of $P_3$. Using the definition of $T$ we get that:
$$
\begin{split} T\begin{pmatrix} 1&0\\0&0 \end{pmatrix}&=1+x+x^2 \\ T\begin{pmatrix} 0&1\\0&0 \end{pmatrix} &=1-x+x^3 \\ T\begin{pmatrix} 0&0\\1&0 \end{pmatrix} &= 1-x+x^3 \\ T\begin{pmatrix} 0&0\\0&1 \end{pmatrix} &=x^2-x^3 \end{split}
$$
Now in the standard basis of $P_3$, $1+x+x^2$ corresponds to the column vector:
$$
\begin{pmatrix} 1\\1\\1\\0 \end{pmatrix}
$$
Combining the column vectors for each of the basis matrices gives the matrix of $T$ in the standard basis:
$$
T=\begin{pmatrix} 1&1&1&0 \\ 1&-1&-1&0 \\ 1&0&0&1 \\0&1&1&-1 \end{pmatrix}
$$
It would be a good exercise for you to do part b). The process will be the same but you will need to write the polynomials you get in terms of the polynomials in the new basis. For example, in the new basis $C$, the polynomial $1+x+x^2$ now corresponds to the vector:
$$
\begin{pmatrix} 1\\0\\0\\0 \end{pmatrix}
$$
</s>