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Showing a set of matrices is a direct sum.

  1. Mar 12, 2013 #1
    Let W1 = {A[itex]\in[/itex] MnXn(R)| A = AT} and W2 = {A[itex]\in[/itex] MnXn(R)| A = -AT}

    Show that MnXn = W1 (+) W2

    where the definition of direct sum is:

    V is the direct sum of W1 and W2 in symbols:

    V = W1 (+) W2 if:

    V = W1 + W2 and
    W1 [itex]\cap[/itex] W2 = {0}


    Attempt:

    I figure I have to show each property individually. So for the first property I tried to do a manipulation:

    AT + (-AT) = (A + (-A))T = 0T

    Then I added A to each side: A + (A + (-A))T = A + 0T

    Did I even show what was needed?
     
  2. jcsd
  3. Mar 12, 2013 #2

    HallsofIvy

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    Staff Emeritus
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    No, you appear to be misunderstanding what "W1= {A[itex]\in[/itex] Mxn|A= AT}" and "W2= {A[itex]\in[/itex] Mnxn|A= -AT[/sub]}" mean. Members of V are of the form A+ B where A is in W1 and B is in W2. You cannot use the same matrix, A, for each.
     
  4. Mar 12, 2013 #3



    How's this:

    Letting A be a matrix from W1 and B be a matrix from W2:

    V = A + B
    = AT+ (-BT)
    = (A+(-B))T

    Can I bring out the -1 and have: (-1)(A+B)T?
     
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