Vector Space Isomorphic

Theorem

• If $V$ is a $\mathbb{F}$ Vector Space
• If $dim(V)=n$
• Then, $V$ is Isomorphic to ${\mathbb{F}}^n$

Idea

• Suppose $\{{\alpha }_1,\dots ,{\alpha }_n\}$ is an Ordered Basis for $V$
• Then, define $T\in L(V,{\mathbb{F}}^n)$ by $T{\alpha }_i=e_i$
• Where $\{e_1,\dots ,e_n\}$ is the Standard Basis for ${\mathbb{F}}^n$