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.
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