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Math Help - vector space

  1. #1
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    vector space

    Question: if {v,w} is a base of a vector space V, then {v+w,v-w} is also a base of the vector space V. Prove?

    need to know the steps to solve or any help would be nice. thanks
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  2. #2
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    Because {v,w} is a basis they are linearly independent.
    So suppose \left( {\exists \alpha  \wedge \beta } \right)\left[ {\alpha (v + w) + \beta (v - w) = 0} \right].
    But that means that \left( {\alpha  + \beta } \right)v + \left( {\alpha  - \beta } \right)w = 0.
    By independence we have \left( {\alpha  + \beta } \right) = 0\,\& \,\left( {\alpha  - \beta } \right) = 0\quad  \Rightarrow \quad \alpha  = 0\,\& \,b = 0

    What does that prove?
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