Linear endomorphism of a vector space of matrices

#linear_algebra

Let $M_{4} (R)$ denote the 16-dimensional real vector space of $4 \times 4$ matrices with real entries, in which the vectors are represented as matrices. Let $T : M_{4} (R) \to M_{4} (R)$ be the linear transformation defined by $T (A) = A - A^{⊤}$ .
(a) Determine the dimension of $\ker (T)$ .
(b) Determine the dimension of $im (T)$ .

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Let $M_4({\bf R})$ denote the 16-dimensional real vector space of $4\times 4$ matrices with real entries, in which the vectors are represented as matrices. Let $T:M_4({\bf R})\to M_4({\bf R})$ be the linear transformation defined by $T(A)=A-A^{\top}$.
\begin{enumerate}[label=\alph*)]
	\item Determine the dimension of $\operatorname{ker}(T)$.
	\item Determine the dimension of $\operatorname{im}(T)$.
\end{enumerate}