Prove that a function is a linear transformation.

Question

Lets say that I have a vector space $A$ and a linear transformation defined as $f : A → A$.

Now I have a function $g : A → A$ defined as $g(a) = bf(a)$ where $a\in A$ and $b \in \mathbb{R}$ is a scalar number.

How would I go about proving whether $g$ is a linear transformation? I'm automatically assuming that it would not be possible to use a specific numerical value for the proof. I understand that for a function to be a linear transformation it has to preserve additivity and scalar multiplication and as $f$ is a linear transformation, it holds certain properties but I'm not sure how to tie it in with the proof for $g$.

Answer 1

Let $u,v\in A$ and $\alpha\in\mathbb{R}$

Then, $$g(u+v)=bf(u+v)=b[f(u)+f(v)]=bf(u)+bf(v)=g(u)+g(v)$$

$$g(\alpha u)=bf(\alpha u)=\alpha bf(u)=\alpha g(u)$$

By this two property, $g$ is linear transformation.