Linear endomorphism of a vector space of matrices
Let
- Determine the dimension of
. - Determine the dimension of
.
View code
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}