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Math Help - Basis Proof

  1. #1
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    Basis Proof

    Let {u,w,v} be a basis for vector space V. Prove that {u+v+w, v+w,w} is also a basis for vector space V
    --------------------
    Ok so the show that this is a basis, it must satisfy being linearly independent and must be able to span any vector spanned by the {u,v,w}.

    So I can see that the vector sum u+v+w cannot be linearly dependent because of the fact that any vector can be written as a sum of these three vectors and at least one non-zero coefficient from {u,v,w} being a basis. It also follows in the same manner the v+w and w are not linear dependent sums. The problem I have is showing that the sum of all three of these is linearly independent. Hmmm...

    As for spanning vector space V, let's say that a_1u+a_2w+a_3v can represent any vector in V by definition. Then a_1u can be written in terms of the "basis" we are trying to prove as a_1(u+v+w)-a_1(v+w), a_2w as simply a_2w, and a_3v as a_3(v+w)-a_3w.

    Look ok minus the missing part which I need help on?
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  2. #2
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    Quote Originally Posted by Jameson View Post
    Let {u,w,v} be a basis for vector space V. Prove that {u+v+w, v+w,w} is also a basis for vector space V
    --------------------
    Ok so the show that this is a basis, it must satisfy being linearly independent and must be able to span any vector spanned by the {u,v,w}.

    So I can see that the vector sum u+v+w cannot be linearly dependent because of the fact that any vector can be written as a sum of these three vectors and at least one non-zero coefficient from {u,v,w} being a basis. It also follows in the same manner the v+w and w are not linear dependent sums. The problem I have is showing that the sum of all three of these is linearly independent. Hmmm...

    As for spanning vector space V, let's say that a_1u+a_2w+a_3v can represent any vector in V by definition. Then a_1u can be written in terms of the "basis" we are trying to prove as a_1(u+v+w)-a_1(v+w), a_2w as simply a_2w, and a_3v as a_3(v+w)-a_3w.

    Look ok minus the missing part which I need help on?
    Your argument for spanning looks solid to me.

    To prove independence, I think this is what you want to do:

    Independent:

    Show that the only solution to \alpha (u + v + w) + \beta (v + w) + \gamma w = 0 is \alpha = \beta = \gamma = 0.

    \alpha (u + v + w) + \beta (v + w) + \gamma w = 0

    \Rightarrow \alpha u + (\alpha + \beta) v + (\alpha + \beta + \gamma ) w = 0.

    Since u, v and w are a basis it follows that

    \alpha = 0 ... (1)

    \alpha + \beta = 0 ... (2)

    \alpha + \beta + \gamma = 0 ... (3)

    The solution to equations (1), (2) and (3) is clearly \alpha = \beta = \gamma = 0 and so independence is proved.
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