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Math Help - Linear Algebra Proof

  1. #1
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    Red face Linear Algebra Proof

    Hi all,

    I need to prove this:

    If The columns of A span Rn, then The equation Ax = b has at least one solution for each b in Rn

    Step by step please....i really want to understand this...thanks
    thanks
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  2. #2
    MHF Contributor
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    Re: Linear Algebra Proof

    If the columns of A span ℝn, it means that for every b ∈ ℝn there exist numbers x1, ..., xn such that x1A1 + ... + xnAn = b, where Ai is the ith column of A. But this exactly means that Ax = b has the solution x = (x1, ..., xn).
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  3. #3
    MHF Contributor

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    Re: Linear Algebra Proof

    That's pretty much what spans means isn't it? x_1A_1+ x_2A_2+ \cdot\cdot\cdot+ x_nA_n is exactly the definition of "Ax". So that is simply saying, "for any b\in R^, there exist x such that Ax= b".
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