A vector space of square matrices

#linear_algebra

Let $M_{n} (R)$ be the vector space of all $n \times n$ matrices with real entries. We say that $A, B \in M_{n} (R)$ commute if $A B = B A$ .

Fix $A \in M_{n} (R)$ . Prove that the set of all matrices in $M_{n} (R)$ that commute with $A$ is a subspace of $M_{n} (R)$ .
Let $A = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}] \in M_{2} (R)$ and let $W \subseteq M_{2} (R)$ be the subspace of all matrices of $M_{2} (R)$ that commute with $A$ . Find a basis of $W$ .

View

L A T E X

code

Let $\operatorname{M}_n({\bf R})$ be the vector space of all $n \times n$ matrices with real entries. We say that $A, B \in \operatorname{M}_n({\bf R})$ commute if $AB = BA$.
\begin{enumerate}[label=\alph*)]
	\item Fix $A \in \operatorname{M}_n({\bf R})$. Prove that the set of all matrices in $\operatorname{M}_n({\bf R})$ that commute with $A$ is a subspace of $\operatorname{M}_n({\bf R})$.
	\item Let $A=\begin{bmatrix} 1 & 1 \\ 1 & 1  \end{bmatrix}\in \operatorname{M}_2({\bf R})$ and let $W\subseteq \operatorname{M}_2({\bf R})$ be the subspace of all matrices of $\operatorname{M}_2({\bf R})$ that commute with $A$. Find a basis of $W$. 
\end{enumerate}