Vector Subspaces

Suppose that V and W are vector subspaces of Rn. If I define:

V + W = {v+w: v belonging to V, w belonging to W}

How can I prove that V+W is also a vector subspace of Rn and ALSO how could verify that (for example)
<(1,0,1), (-1,0,1)> + <(3,2,1)> = R3

Thanks

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Follow up from OTA on Student Comment:
To answer your questions in the comment:

Yes, it works with any square matrix.

I have in my old algebra text a theorem (Elementary Linear Algebra with Applications by W. Keith Nicholson):
The ...

Vector subspaces are investigated. The solution is detailed and well presented.

$2.19