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Subspace question

  • Thread starter yanjt
  • Start date
  • #1
14
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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!
 

Answers and Replies

  • #2
392
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Not closed under matrix addition.
 
  • #3
726
1
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.
 
  • #4
matt grime
Science Advisor
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In general, the relation det(a + b) = det(a) + det(b) doesn't hold.
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).
 
  • #5
14
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Hi, I do not understand why it is not closed under matrix addition.It is still a Mn,n matrix isn't it?
 
  • #6
392
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Find two matrices whose det is 0 but whose sum has nonzero det.
 
  • #7
matt grime
Science Advisor
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Hi, I do not understand why it is not closed under matrix addition.It is still a Mn,n matrix isn't it?
But that isn't the only criterion you had to have the sum satisfy, is it?
 

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