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Subspace question

  1. Apr 17, 2009 #1
    Hi,
    I would like to know why the set of all n*n matrix whose determinant is zero is not a subspace of Mn,n .Can anyone explain the reason for me?

    Thanks!
     
  2. jcsd
  3. Apr 17, 2009 #2
    Not closed under matrix addition.
     
  4. Apr 17, 2009 #3
    In general, the relation det(a + b) = det(a) + det(b) doesn't hold. Try to think of an example where det(a) = det(b) = 0 but det (a + b) is not equal to zero.
     
  5. Apr 17, 2009 #4

    matt grime

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    True, but not the point. If det were a linear map then its kernel is a subspace. Its kernel not being a subspace implies det is not a linear map. But that does not mean that its kernel is not a subspace (exercise: find an example of a map f from a vector space to the base field so that f is not linear, but the set of x such that f(x) is zero is a subspace).
     
  6. Apr 18, 2009 #5
    Hi, I do not understand why it is not closed under matrix addition.It is still a Mn,n matrix isn't it?
     
  7. Apr 18, 2009 #6
    Find two matrices whose det is 0 but whose sum has nonzero det.
     
  8. Apr 18, 2009 #7

    matt grime

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    But that isn't the only criterion you had to have the sum satisfy, is it?
     
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