Elementary Linear Algebra 7th Edition

The set $W$ is a vector subspace $M_{n,n}$.
The set $W$ of all $n \times n$ matrices that commute with a given matrix $B$ is a vector subspace $M_{n,n}$. Indeed; Let $A$ and $C$ be two matrices that commute with the matrix $B$. Now, $(A+C)B=AB+AC=BA+BC=B(A+C)$ and also for any $c\in R$ $(cA)B=B(cA)$, hence $W$ is a vector subspace $M_{n,n}$.