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Math Help - exams soon....Prove that {u+v,u-v} is a basis for S

  1. #1
    n22
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    exams soon....Prove that {u+v,u-v} is a basis for S

    Dear Smart people,
    Please help me answer this question.
    Let u and v be linearly independent vectors in a vector space V.
    Let S=Span{u,v}
    Prove that {u+v,u-v} is a basis for S
    thanks!
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  2. #2
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    Re: exams soon....Prove that {u+v,u-v} is a basis for S

    Quote Originally Posted by n22 View Post
    Please help me answer this question.
    Let u and v be linearly independent vectors in a vector space V.
    Let S=Span{u,v}
    Prove that {u+v,u-v} is a basis for S.
    You must show that u+v~\&~u-v are linearly independent vectors.

    You must show that if w\in S then \exists~\alpha~\&~\beta such that w=\alpha(u+v)+\beta(u-v)
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    Re: exams soon....Prove that {u+v,u-v} is a basis for S

    First, since u and v are linearly independent vectors that span S, they are a basis for S.
    To show that {u- v, u+ v} is a basis for S, you must show that they also are independent and span S.

    So you must show that if a(u- v)+ b(u+ v)= 0 for numbers, a and b, then a= 0 and b= 0. You can do that by combing multiples of u and combining multiples of v and using the fact that u and v are independent.

    To show that u- v and u+ v span S you must show that any vector in S can be written as a linear combination of u- v and u+ v. Of course, because u and v span S, any vector in S can be written as au+ bv.
    Thanks from n22
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