# Linear Transformation Formal Proof

By definition of a linear transformation, the following statement must be true, but how does one formally prove it?

Let $f : V \rightarrow V$ be a linear transformation on vector space $V$, and $g : V \rightarrow V$ be a function such that for every $v \in V$

$af(v) = g(v),\> a \in R$.

Prove that $g$ is also a linear transformation.

For $v_1,v_2\in V$ $$g(v_1+v_2)=af(v_1+v_2)=a(f(v_1)+f(v_2))=g(v_1)+g(v_2)$$ similarly $$g(cv)=af(cv)=acf(v)=cg(v)$$
Here we use the fact that $f$ is a linear transform and definition of linear transform.