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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?

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

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