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Linear Algebra: The vector space R and Rank

  1. Nov 19, 2005 #1
    Two m x n matrices A and B are called EQUIVALENT (writen A ~e B if there exist invertible matracies U and V (sizes m x m and n x n) such that A = UBV
    a) prove the following properties of equivalnce
    i) A ~e A for all m x n matracies A
    ii) If A ~e B, then B ~e A
    iii) A ~e B and B~e C, then A~e C

    --------------

    Lets just do part i) for now...

    It seems ovicus that "A ~e A for all m x n matracies A"... but.. heres how i would do it

    A = m x n
    U = m x m
    V = n x n

    So

    A = UAV
    A = (UA)V UA = (m X m)(m X n) = (m X n)
    A = (UA)(V) UAV = (m X n)(n x n) = (m x n)
    A = A

    How does that sound? Is that how you prove this? Or do i have the wrong idea?

    Please help

    thanks
     
  2. jcsd
  3. Nov 20, 2005 #2

    matt grime

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    Science Advisor
    Homework Helper

    What is your idea? How can A=m x n? That makes no sense. A is an mxn matrix, but not equal to m x n, whatever that means.
    Find U and V, invertible matrices such that A=UAV, that's all you need to do.
    You can't start by setting A=UAV, as you do, since that is assuming the answer. I don't even know what your argument is trying to do since you haven't used any words to explain any of your steps.
    As for the other two: if you actually write out what you need to show, and what you are given, then it is a simple manipulation of matrices.
     
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