1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    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

  2. jcsd
  3. Nov 20, 2005 #2

    matt grime

    User Avatar
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook