# sub and vectorspace

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• May 26th 2010, 05:41 AM
Kredje
sub and vectorspace
Hi i encountered this problem while doing my homework, if anyone can give me the answer i would be very happy.

Let H be an n-dimensional subspace of an n-dimensional vectorspace V. Show that H = V

Answers in dutch are welcome aswell, thanks in advance.
• May 26th 2010, 05:54 AM
HallsofIvy
Quote:

Originally Posted by Kredje
Hi i encountered this problem while doing my homework, if anyone can give me the answer i would be very happy.

Let H be an n-dimensional subspace of an n-dimensional vectorspace V. Show that H = V

Answers in dutch are welcome aswell, thanks in advance.

Since you are given that H is a subspace of V, to show that H= V, you need only show that any vector in V is also in H.

Let v be any vector in V and select a basis, $\{h_1, h_2, \cdot\cdot\cdot,h_n\}$ for H. If v were not in H, then it could not be written as a linear combination of those basis vectors and so $\{h_1, h_2, \cdot\cdot\cdot, h_n, v\}$ would be a linearly independent set of n+ 1 vectors in V. Do you see why they are linearly independent and why that contradicts the fact that V has dimension n?
• May 26th 2010, 06:25 AM
Kredje
HallsOfIvy thanks for the quick respons. I think i understand now :).