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Finding the Dimension and Basis of the Matrix Vector space

  1. Oct 14, 2012 #1
    1. The problem statement, all variables and given/known data
    The set K of 2 × 2 real matrices of the form [a b, -b a] form a field with the usual operations.
    It should be clear to you that M22(R) is a vector space over K. What is the dimension of M22(R) over K? Justify your answer by displaying a basis and proving that the set displayed is actually a basis.


    2. Relevant equations



    3. The attempt at a solution

    I don't think there can be a basis over the field K, because no linear combination of the matrices [a b, -b a] with any M22 can form, say [1 0, 0 0]. Which would be in M22. Any help would be greatly appreciated. Thanks!
     
  2. jcsd
  3. Oct 16, 2012 #2
    Could you do a combination say (0 a, 0 -b) (0 -b, 0 a), (a 0, b 0), (-b 0, a 0)? Could this form a basis? Or does it have to be real integers? thanks
     
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