Can you guys help me with this question?
Let $T:{M_{2\times2}} \to {P_2}$ be defined by $$T\begin{pmatrix} a&b\\c&d \end{pmatrix}=(a+b-c-d)t^2+ (c+d)t+ (a+b). $$ Find the matrix of T with respect to the standard bases for ${M_{2\times 2}}$ and ${P_2}$.
The standard bases for ${P_2}$ is $$\left\{ {1,t,{t^2}} \right\}$$ and for ${M_{2×2}}$ is $$\left\{ \begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix},
\begin{pmatrix}
0 & 1 \\
0 & 0
\end{pmatrix},
\begin{pmatrix}
0 & 0 \\
1 & 0
\end{pmatrix},
\begin{pmatrix}
0 & 0 \\
0 & 1
\end{pmatrix} \right\}$$
Best Answer
The matrix of a linear transformation comes from expressing each of the basis elements for the domain in terms of basis elements for the range upon applying the transformation.
For example,
$$T \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} = 1\cdot 1 + 0\cdot t + 1 \cdot t^2.$$ Thus the first column of the matrix for $T$ with respect to these bases will be $$\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}.$$ Repeat for the three remaining basis elements of $M_{2 \times 2}$.
Solution: