Hi all.

I am a beginner to real maths and am looking at a problem from a book by Loomis and Sternberg.

Let $\displaystyle T$ be the linear transformation from $\displaystyle \mathbb{R}^B$ to $\displaystyle \mathbb{R}^A$ such that $\displaystyle T(f) = f \circ \varphi $.

I need to show that $\displaystyle T$ is an isomorphism if $\displaystyle \varphi$ is a bijection by showing that:

(a) $\displaystyle \varphi$ injective -> $\displaystyle T$ surjective;

(b) $\displaystyle \varphi$ surjective -> $\displaystyle T$ injective.

I'm not really sure where to start, and I can't even see the link between the properties of the 2 mappings; any hints would be greatly appreciated.

Thanks