Pigeonhole Principle

Definition

Let $A,B$ be sets and $|A|=|B|$, and $f:A\to B$. Then $f$ is Surjective $\iff$ $f$ is Injective $\iff$ $f$ is Bijective

Lemma

Let $L:V\to W$ be a linear map, and suppose that $\dim (V)=\dim (W)$. Then $L$ is Epimorphism $\iff$ $L$ is Monomorphism $\iff$ $L$ is Isomomorphism

Proof

Epimorphism $\Rightarrow$ Monomorphism

Let $\dim (V)=\dim (W)=n$ If $L$ is Monomorphism $\Rightarrow \dim (L(V))=\mathrm{im}(L)=n$ by the definition or Monomorphism $\Rightarrow \dim (\mathrm{Ker}L)=n-n=0\iff \mathrm{Ker}L=\{0\in V\}$ by the Dimension Theorem So, $\forall {\mathbf{v}}_1,{\mathbf{v}}_2\in V:L({\mathbf{v}}_1)=L({\mathbf{v}}_2)\Rightarrow L({\mathbf{v}}_1)-L({\mathbf{v}}_2)=0\Rightarrow L({\mathbf{v}}_1-{\mathbf{v}}_2)=0\in W\Rightarrow {\mathbf{v}}_1-{\mathbf{v}}_2=0$ by the $\mathrm{Ker}L=\{0\}$ $\therefore {\mathbf{v}}_1={\mathbf{v}}_2$

Monomorphism $\Rightarrow$ Epimorphism

$\dim (\mathrm{Ker}L)=0\Rightarrow \dim (L(V))=n$ by the Dimension Theorem $L(V)\subset W$ and $\dim (L(V))=\dim (W)$ $\therefore L(V)=W=\mathrm{im}L$

How to prove this statement? Let V be a vector space and W \subset V be given, If dim V = dim W, then V = W