Question

Let $M_{2 \times 2}$ be the vector space of all $2 \times 2$ matrices, and define $T: M_{2 \times 2} \rightarrow M_{2 \times 2}$ by $T(A)=A+A^T$, where $A=\left[[tex][tex]\begin{array}{ll}a & b \\ c & d\end{array}\right]$[/tex][/tex]
a. Show that $T$ is a linear transformation.
b. Let $B$ be any element of $M_{2 \times 2}$ such that $B^T=B$. Find an $A$ in $M_{2 \times 2}$ such that $T(A)=B$.
c. Show that the range of $T$ is the set of $B$ in $M_{2 \times 2}$ with the property that $B^T=B$.
d. Describe the kemel of $T$.