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Math Help - I need help to prove this in Linear Algebra

  1. #1
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    I need help to prove this in Linear Algebra

    If The linear transformation x Ax maps Rn onto Rn.

    then
    The columns of A span Rn.

    I proved the other way...but in order for this proof to be complete...I need to prove this..step by step please...it is important that I do understand the proof as well....thanks in advance
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  2. #2
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    Re: I need help to prove this in Linear Algebra

    This is similar to the other problem. Given x = (x1, ..., xn)T, where the superscript T means transpose, i.e., this is a column vector, Ax = x1A1 + ... + xnAn where Ai is the ith column of A and this is the sum of column vectors. The fact that x ↦ Ax is onto means that for every b ∈ ℝn there exist an x such that Ax = b. But this also means that the columns of A span ℝn.
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  3. #3
    Senior Member jakncoke's Avatar
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    Re: I need help to prove this in Linear Algebra

    Doesn't it follow from the fact that the linear transformation is "onto" ?

    Means that any vector in \mathbb{R}^{n} can be gotten by doing Ax, where x is the coordinate written under some basis B. That itself means that any vector in \mathbb{R}^n can be written in terms of c_1 v_1 + ... +c_n v_n where v_1,..v_n are the columns of A and c_1,..c_n is a coordinate under some basis B. thus the columns of A span \mathbb{R}^n
    Last edited by jakncoke; March 19th 2013 at 01:39 PM.
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