Injection, Surjection and Bijection in

Question

I came across this claimed theorem:

Let $V$ and $W$ be finite-dimensional vector-spaces of equal dimension. Let $L: V \rightarrow W $ be a linear function.

Then the following statements are equivalent:

• $L$ is injective
• $L$ is surjective
• $L$ is a bijection

Can someone give me a short summary why this is ?

Answer 1

Hint:

Use the rank-nullity formula and remember ‘$L$ injective’ is equivalent to ‘$\ker L=\{0\}$’, ‘$L$ surjective’ is equivalent to ‘$\dim (\operatorname{Im}L)=\dim W$’.