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Math Help - Algebra1

  1. #1
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    Exclamation Algebra1

    Let A be an m x n matrix, show that the following are equivalent:
    (a) A has a right inverse, that is, there exists an n x m matrix C such that AC=I.
    (b) The system Ax=b has at least one solution x for each b in Rm.
    (c) The columns of A span Rm.

    Show that any one of (a),(b) or (c) implies that m<=n.


    Thanks very much.
    Last edited by CaptainBlack; May 11th 2006 at 09:08 PM.
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  2. #2
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    Quote Originally Posted by suedenation
    Let A be an m x n matrix, show that the following are equivalent:
    (a) A has a right inverse, that is, there exists an n x m matrix C such that AC=I.
    (b) The system Ax=b has at least one solution x for each b in Rm.
    (c) The columns of A span Rm.

    Show that any one of (a),(b) or (c) implies that m<=n.


    Thanks very much.
    This is a one of those proof that is 'natsy' to write out and relatively easy. It is partically a definition of what vector spaces are. Anywars this is called the "Fundamental Theorem of Existence of a Solution for a Linear System" (long name), maybe, if you want write out for you?
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