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Math Help - Elem. proof help?

  1. #1
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    Elem. proof help?

    Hi, just reviewing some linear algebra this summer. Came across this in one of my exercise books, and was looking for a proof.

    Let A be an m x n matrix with rank m. Prove that there exists an n x m matrix B such that AB = I(m).

    [That last part reads "AB equals I sub m."]
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Consider the linear transormation R_A : \mathbb{R}^m \rightarrow \mathbb{R}^n : v \mapsto vA. By definition of the rank, R_A is an injection, hence is left-invertible; hence there exists a transformation T:\mathbb{R}^n \rightarrow \mathbb{R}^m such that T\circ R_A=1. Writing T=R_B we have R_B\circ R_A=1; but R_B\circ R_A=R_{AB} so AB=I_m.
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