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Math Help - Vectors x,y in V (this seems to easy)

  1. #1
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    Vectors x,y in V (this seems to easy)

    V is a vector space over a field F and A is a subspace. x,y\in V have two lin. ind. combinations which are in A. Show x,y\in A

    Let \lambda_i\in F

    x=\lambda_1v_1+\cdots +\lambda_nv_n\in A

    y=\lambda_1v_1+\cdots +\lambda_mvm\in A

    Since x = the linear combinations in A, x is in A. Is it really this simple?
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    Re: Vectors x,y in V (this seems to easy)

    Quote Originally Posted by dwsmith View Post
    V is a vector space over a field F and A is a subspace. x,y\in V have two lin. ind. combinations which are in A. Show x,y\in A

    Let \lambda_i\in F

    x=\lambda_1v_1+\cdots +\lambda_nv_n\in A

    y=\lambda_1v_1+\cdots +\lambda_mvm\in A

    Since x = the linear combinations in A, x is in A. Is it really this simple?
    Hi dwsmith,

    I think you have misunderstood the question. x,y\in V have two lin. ind. combinations which are in A, means that,

    \lambda_1 x+\lambda_2 y\in A\mbox{ and }\lambda_1 x+\lambda_2 y\in A\mbox{ for some }\lambda_1,\lambda_2,\lambda_3,\lambda_4\in F
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    Re: Vectors x,y in V (this seems to easy)

    Quote Originally Posted by Sudharaka View Post
    Hi dwsmith,

    I think you have misunderstood the question. x,y\in V have two lin. ind. combinations which are in A, means that,

    \lambda_1 x+\lambda_2 y\in A\mbox{ and }\lambda_1 x+\lambda_2 y\in A\mbox{ for some }\lambda_1,\lambda_2,\lambda_3,\lambda_4\in F
    Sudharaka clearly meant \lambda_3x+ \lambda_4y \in A for the second.

    However, you still have to assume that "linear combinations" means "none of the coefficients are 0" which is not standard. Otherwise you would have 0x+ 1y\in A and 0x+ 2y\in A which are true for any y in A whether x is in A or not.
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