Question

Let B be a 4 × 4 diagonal matrix, and let WB be the set of 4 × 4 matrices which
commute with B (i.e., those matrices A so that AB = BA)

Check that WB is a a subspace of M4×4(R)

Answer 1

I'm pretty bad at strong proofs for this kind of thing, but\[AB+CB=BA+BC=(A+C)B\implies B(A+C)\in\ W\]so it seems closed under addition

Answer 2

Hardly much to show for scalar multiplication\[c(AB)=c(BA)=cAB=cBA\in W\]

Answer 3

and the zero vector is quite commutative... so you can do that part yourself