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Linear Algebra Questions

  1. Oct 22, 2005 #1
    I am having trouble with the following questions
    1) Let U be such that AU=0 implies that A=0. If AU=BU, show that A=B.
    So far, I did this, but it doesnt seem right to me.
    To Show:
    Therefore A = B.
    2)If A=
    [a b]
    [c d] (2x2 matrix, sorry not to sure on how to place them here)

    where a is not equal to 0, show that A factors in the form A =
    [1 0][y z]
    [x 1][0 w] (those are two 2x2 matricies multiplied together)

    Im not too sure on how to start on this question at all.

    Thanks in advance
  2. jcsd
  3. Oct 22, 2005 #2


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    Gold Member

    Your mistake in the first one is in assuming A=0. This is not necessarily the case. You want to show that [itex]AU=BU \rightarrow A=B[/itex], or equivalently [itex]AU-BU=0 \rightarrow A=B[/itex]. You never said what kind of objects A, B and U are, but presumably they have some sort of linearity property, correct? If this is the case, then it is easy to rearange the equation to get it to say something times U = 0. You are told in the problem that this implies that something is zero. Take it from there.
  4. Oct 22, 2005 #3
    Sorry, A, U, and B are matricies. Im not too sure I understand the part when you said A may not be 0. What does it mean when it says "implies that A = 0"?

  5. Oct 22, 2005 #4


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    Gold Member

    When they say "AU=0 implies that A=0" they mean that if AU=0, then A=0 where A is any matrix that can be multiplied by U. Don't get confused by the fact that they later use the same letter when they say "If AU=BU, show that A=B". When they say A, they mean any matrix.
  6. Oct 23, 2005 #5


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    Staff Emeritus
    Science Advisor

    If AU= BU then AU-BU= 0. That's what you need to use "If AU= 0,...".
  7. Oct 23, 2005 #6
    Great! Im pretty sure I understand it now. Thanks a lot
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