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Math Help - linear algebra proof

  1. #1
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    linear algebra proof

    Suppose that V is a vector space and that S = {u1,,un} is one basis for V. Prove that the set T = {u1,,un-1} is a linearly independent set but that it will not span V.

    I think it is easy to prove that T is linearly independent, but how to conclude with it will not span V?

    Thanks in advance!
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  2. #2
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    Quote Originally Posted by gavin1989 View Post
    Suppose that V is a vector space and that S = {u1,,un} is one basis for V. Prove that the set T = {u1,,un-1} is a linearly independent set but that it will not span V.

    I think it is easy to prove that T is linearly independent, but how to conclude with it will not span V?

    Thanks in advance!

    Your told that the vectors in T are independent, just prove that you cannot construct Un form the elements in T.

    Bobak
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  3. #3
    Super Member Showcase_22's Avatar
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    V has a dimension of n. This means that you need n linearly independent vectors to form a basis of V.

    However, in the new basis you only have n-1 vectors. Even if these vectors are linearly independent, they cannot span the space because you only have n-1 of them when you need n.

    This was a theorem in my lecture notes, you might be able to find something similar in yours.
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