U = {A ∈ R^n | BA = 0(nxn)} , prove using subspace test

Question

U = {A ∈ $R$ | BA = $0(nxn)$} , prove using subspace test.

I'm going to guess that since BA = $0(nxn)$ then A = $0(nxn)$, making B = $0(nxn)$. Therefore $0(nxn)$ $R$.

Not sure how to prove it is closed under addtion or scalar multiplication.

Answer 1

Since $$B0=0$$

$U$ is non-empty.

If $A_1, A_2 \in U$, we have $BA_1=0$ and $BA_2=0$

hence $$B(A_1+A_2)=BA_1+BA_2=0$$

If $k \in \mathbb{R}$, then $B(kA_1)=k(BA_1)=0$