$V$ finite-dimensional vector space and isomorphic to $\mathbb{R}^n$?

Question

If $V$ is a finite-dimensional vector space, does it mean that $V$ is also isomorphic to $\mathbb{R}^n$ for some $n$? I am having a hard time trying to picture this. I was wondering if someone could explain this to me.

Answer 1

Let $V$ have dimension $n$ over $\Bbb R$, say with basis $\{v_1,\dots,v_n\}$. Define a map $f:V\to\Bbb R^n$ by

$$f(a_1v_1+\dots+a_nv_n)=(a_1,\dots,a_n).$$

I encourage you to check for yourself that this is linear, injective and surjective.