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How to show the direct sum of two matrices?

  1. Feb 17, 2009 #1
    1. The problem statement, all variables and given/known data
    Let k be a field, V = Mat2x2(k), U:={[a, b], [-b, a] a, b E k} and W:={[a, b], [b, -a] a, b E k}. Show that V is the direct sum of U and W.


    2. Relevant equations



    3. The attempt at a solution

    Add the matrix for U to the matrix of W. Values in that matrix still exist in k, so V = U + W. Is that the right reasoning? How do you show the intersection of two matrices is 0? Very confused.
     
  2. jcsd
  3. Feb 17, 2009 #2

    Dick

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    Homework Helper

    Write W={[c, d], [d, -c] c, d E k} to make it clear there are four independent parameters here. Equate an element of U with an element of W and show a=0, b=0, c=0 and d=0 from the equations you get. Then you have to show any element of V can be written as a sum of an element from U and an element from W.
     
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