Prove that $\mathcal{AB}$ is linear operator if $\mathcal{A}$ and $\mathcal{B}$ are linear operators.

Question

It is fairly easy to determine whether $\mathcal{AB}$ is linear when we know $\mathcal{A}$ and $\mathcal{B}$ (for example, $\mathcal{Ax}=(2x_1, 3x_2-x_1)$ and $\mathcal{B}$ is something similar). But how to prove it in general?

Answer 1

You need to prove two things:

1. For all vectors $u,v$ in the domain of $\mathcal{B}$, $\mathcal{AB}(u+v)=\mathcal{AB} (u)+\mathcal{AB}(v)$
   
2. For all vectors $u$ in the domain of $\mathcal{B}$, and all scalars $k$ in your common underlying field, $\mathcal{AB}(ku)=k\mathcal{AB}(u)$.

The technique is to use repeatedly these same properties for $\mathcal{A}$ and $\mathcal{B}$ separately. The second thing is proved as: $\mathcal{AB}(ku)=\mathcal{A}(\mathcal{B}(ku))=\mathcal{A}(k\mathcal{B}(u))=k\mathcal{A}(\mathcal{B}(u))=k\mathcal{AB}(u)$