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.
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.
Originally Posted by Kredje
Let v be any vector in V and select a basis, for H. If v were not in H, then it could not be written as a linear combination of those basis vectors and so 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?
HallsOfIvy thanks for the quick respons. I think i understand now :).