Elementary Linear Algebra

Discussion in 'Undergraduate Math' started by Ladnor Geissinger, Apr 12, 2004.

  1. Recently there have been many notes about problems involving linear
    independence, spanning, and row-reducing a matrix. Perhaps some
    readers will find the following way of thinking about such problems
    useful.

    Suppose that we have matrices where: A is nXn, B is nXp, C is pX1, and
    A is invertible. For example, A might be the product of elementary
    matrics that you could accumulate while row reducing B, i.e. U=(A*B)
    is row-reduced. And C might represent a linear dependence relation
    among the columns of B, i.e. B*C=0.
    By the associative property, (A*B)*C = A*(B*C) for any compatible
    A,B,C.
    So, when A is invertible, C represents a linear relation on the
    columns of B iff C also represents a linear relation on the columns of
    U=(A*B).
    So, for example, columns 1,2,4,7 of B are linearly indep. iff columns
    1,2,4,7 of U are lin. indep., which would be obvious if 1,2,4,7
    happened to be pivot columns.
     
    Ladnor Geissinger, Apr 12, 2004
    #1
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