[Math] How to prove that if T is surjective, then T is an isomorphism.

Question

Let V be a finite-dimensional vector space, and let T : V → V be a linear transformation. Prove that if T is surjective, then T is an isomorphism.

My attempt: By definition, surjection is that every element on the codomain is covered. Therefore, if every element on T is covered, and since it is mapping to itself, it must be an isomorphism (also by definition).

Answer 1

Establish the following facts:

1. (Rank-Nullity Theorem) $\dim \ker T + \dim \operatorname{im} T = \dim V$. For this you need to explicitly define a basis.

2. Since $T$ is surjective, $\dim \operatorname{im} T = \dim V$, so $\dim\ker T = 0$

3. If $\dim \ker T = 0$, then $T$ is injective. (Show that $\dim\ker T = 0 \iff \ker T = \{0\}$)