Using the dimension formula to prove isomorphism

Question

Let $V$ be a finite-dimensional vector space and $T: V\rightarrow V $. $T$ is a linear transformation. Use the dimension formula to prove that if $T$ is injective, it must also be surjective; if T is surjective, it must also be injective.

Answer 1

Let $T:V\to V$ be a linear map on a finite-dimensional vector space $V$. Recall that

• $T$ is injective if and only if $\dim\ker T=0$
• $T$ is surjective if and only if $\dim\DeclareMathOperator{image}{image}\image T=\dim V$

The Rank-Nullity Theorem states that the equality $$ \dim\ker T+\dim\image T=\dim V $$ always holds. Can you combine the above to prove your result?

Answer 2

We have these equivalences (using $\dim \ker T+\operatorname{rank}T=\dim V$)

$$T\;\text{is injective}\iff\ker T=\{0\}\iff\dim\ker T=0\\\iff\operatorname{rank}T=\dim V\iff T \;\text{is surjective}$$