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)

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)

turingtest · Answer

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

turingtest · Answer

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

turingtest · Answer

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