Prove that the sum is also linear application

Question

Be two linear applications $f, g : \mathbb R^m \rightarrow \mathbb R^n$ . Show that the sum $f + g : \mathbb R^m \rightarrow \mathbb R^n$, defined as:

$$(f + g)(v) = f(v) + g(v), \forall v \in \mathbb R^m,$$

It is also a linear application.

Answer 1

Let check that the linearity properties hold

• $(f + g)(v_1+v_2)=(f + g)(v_1)+(f + g)(v_2)$
• $(f + g)(cv)=c(f + g)(v)$