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Math Help - Bases for row and column spaces

  1. #1
    MHF Contributor alexmahone's Avatar
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    Bases for row and column spaces

    Without multiplying matrices, find bases for the row and column spaces of A:

    A=\left[ {\begin{array}{cc}1 & 2  \\4 & 5  \\2 & 7  \\ \end{array} } \right]\left[ {\begin{array}{ccc}3 & 0 & 3  \\1 & 1 & 2  \\\end{array} } \right]

    How do you know from these shapes that A cannot be invertible?
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Upon inspection, I find that a basis for the row space of A is (3,0,3) and (1,1,2) while a basis for the column space is (1,4,2) and (2,5,7). Although I'm not sure why...
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    Quote Originally Posted by alexmahone View Post
    Without multiplying matrices, find bases for the row and column spaces of A:

    A=\left[ {\begin{array}{cc}1 & 2  \\4 & 5  \\2 & 7  \\ \end{array} } \right]\left[ {\begin{array}{ccc}3 & 0 & 3  \\1 & 1 & 2  \\\end{array} } \right]
    If we write A= BC, then C maps all of R^3 into a subspace of R^2 which can, of course, have dimension no larger than 2. B then maps that subspace into a subspace of R^3 which can have dimension no larger than 2. Since A= BC maps R^3 into a two dimensional subspace of R^3, it cannot be invertible.

    How do you know from these shapes that A cannot be invertible?
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