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Prove Not a Vector Space

  1. Oct 18, 2012 #1
    1. The problem statement, all variables and given/known data
    Let V= set of 2x2 matrices with the normal addition, but where multiplication is defined as: β#A=β(A^T) where A^T is the transpose of A.


    2. Relevant equations
    The axiom about 1#A=A


    3. The attempt at a solution
    I think that because you can show that not ALL matrices satisfy A=A^T, you can't have a vector space since the multiplication by 1 doesn't hold up.

    But then I'm wondering whether I'm assuming that the multiplicative identity should be the "normal" 1 (ie that 1 is just the scalar 1 in a normal R^n vector space).

    How do you prove a multiplicative identity absolutely does NOT exist?
     
  2. jcsd
  3. Oct 18, 2012 #2

    Zondrina

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    So are you trying to prove that V is a vector space if it obeys the normal matrix addition, but has a unique multiplication scalar multiplication defined as # which is sort of a mapping from A to At?

    Your notation is a bit confusing to me.
     
  4. Oct 18, 2012 #3

    Dick

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    If there were a scalar β that was a multiplicative identity it would have to satisfy β(A^T)=A for all matrices A. Show there isn't.
     
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