sub and vectorspace

May 2010
2
0
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.
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
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, \(\displaystyle \{h_1, h_2, \cdot\cdot\cdot,h_n\}\) for H. If v were not in H, then it could not be written as a linear combination of those basis vectors and so \(\displaystyle \{h_1, h_2, \cdot\cdot\cdot, h_n, v\}\) 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?
 
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May 2010
2
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HallsOfIvy thanks for the quick respons. I think i understand now :).