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Math Help - sub and vectorspace

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by Kredje View Post
    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?
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  3. #3
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    HallsOfIvy thanks for the quick respons. I think i understand now .
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