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


n-dimensional vector space?

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

  • $\begingroup$ 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

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


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.