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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.

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    $\begingroup$ Just map the finite basis set to any basis set of $\mathbb{R}^n$ for some $n$ and extend linearly to whole space. This map then should be an isomorphism. we can do this becase we know the space is finite dimensional. $\endgroup$
    – user160738
    Mar 3 '17 at 18:46
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    $\begingroup$ If it's a finite-dimensional real vector space, then yes. $\endgroup$
    – Bobbie D
    Mar 3 '17 at 18:46
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    $\begingroup$ The answer is no. A simple counter example is the vector space $\mathbb{F}_2^n$ where $\mathbb{F}_2 = \mathbb{Z}/2\mathbb{Z}$. $\endgroup$ Mar 3 '17 at 19:14
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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.

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