# How would you explain this vector space isomorphism theorem?

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?

• It means "vector space", but if one is concentrating on vector spaces one often just says "space". Have you heard of bases? Jun 27 '18 at 10:08
• @ancientmathematician yes, i have heard of vector bases Jun 27 '18 at 10:13

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.