How would you explain this vector space isomorphism theorem?

Question

Every n-dimensional space $(V, F)$ is isomorphic to the vector space $(F^n, F)$ where:

$F^n=\{ (\alpha_1,....\alpha_n), \forall i \in \{1,2,3,...n\} \alpha_i \in F \}$

Should it say

n-dimensional space

or

n-dimensional vector space?

Also, why is it isomorphic to this n-dimensional vector space? How would you explain this?

Answer 1

You should say :"n-dimensional vector space".

Let $v_1,...,v_n$ be a basis of $V$. Then each $v \in V$ has a unique representation

$v=a_1v_1+...+a_nv_n$

with $a_1,...,a_n \in F$.

Then define $T:V \to F^n$ by $T(v)=(a_1,...,a_n)$ and show that $T$ is an isomorphism.