linear algebra - Prove that the set of commuting matrices is a vector space

Prove that the set of real commuting matrices with the matrix $A= \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}$ is a vector space relative to standard operations on matrices. Find the dimension and a basis for that space.

Question: Is it necessary to check the subtraction for commuting matrices.

What are the steps for proving the given statement?

Answer

You just have to show it is a subspace of the vector space of $n\times n$ matrices.

Clearly the zero matrix commutes with $M$.

Supose $A$ and $B$ commute with $M$. then $(A+B)M=AM+BM=MA+MB=M(A+B)$.

We also have $(cA)M=c(AM)=c(MA)=M(cA)$, so the matrices that commute with $M$ contain the zero matrix and are closed under addition and scalar multiplication, therefore they form a subspace of the vector space of matrices as desired.