<s>[INST] Confusion about a Linear Transformation question. Let $\beta := [M_1, M_2, M_3, M_4]$ be the ordered basis of $R^{2×2}$ defined by:
$$ M_1 := \begin{pmatrix}
1 & 0\\
0 & 0 \end{pmatrix}, 
M_2 := \begin{pmatrix}
0 & 1\\
0 & 0 \end{pmatrix},
M_3 := \begin{pmatrix}
0 & 0\\
1 & 0 \end{pmatrix},
M_4 := \begin{pmatrix}
0 & 0\\
0 & 1 \end{pmatrix}
$$
Define the linear transformation
$L := A → (A + A^T)$.
a) What is $L \left( \begin{pmatrix}
a & b\\
c & d \end{pmatrix} \right)$?
b) Find the matrix representation of $L$ with respect to the basis $\beta$.
c) Find all solutions to $L(A) = \begin{pmatrix}
4 & 0\\
0 & -6 \end{pmatrix}$.
Does the following look correct?
$L \left( \begin{pmatrix}
a & b\\
c & d \end{pmatrix} \right)$ = $ \begin{pmatrix}
2a & b + c\\
b+ c & 2d \end{pmatrix}$
$Ker(L) = span\{\begin{pmatrix}
0 & c\\
-c & 0 \end{pmatrix}\}$
$range(L) = span\{ \begin{pmatrix}
a & 0\\
0 & 0 \end{pmatrix}, \begin{pmatrix}
0 & s\\
s & 0 \end{pmatrix}, \begin{pmatrix}
0 & 0\\
0 & d \end{pmatrix}\}$ where s denotes the quantity $c + d$
Matrix Representation: $\begin{pmatrix}
2 & 0 & 0 & 0\\
0 & 1 & 1 & 0\\
0 & 1 & 1 & 0\\
0 & 0 & 0 & 2\end{pmatrix} \begin{pmatrix}
a\\
b\\
c\\
d\end{pmatrix}=\begin{pmatrix}
2a\\
b + c\\
b + c\\
2d\end{pmatrix}$
Thanks for the help!
 [/INST] $$  \begin{pmatrix}
a & b\\
c & d \end{pmatrix}=a.\begin{pmatrix}
1 & 0\\
0 & 0 \end{pmatrix}+b.\begin{pmatrix}
0 & 1\\
0 & 0 \end{pmatrix}+c.\begin{pmatrix}
0 & 0\\
1 & 0 \end{pmatrix}+d.\begin{pmatrix}
0 & 0\\
0 & 1 \end{pmatrix}$$
Now $L$ is linear and then apply $L$ to each of these four matrices given $L(A)=A+A^T$
</s>