Exploring Subspaces in Matrices: Symmetric vs. Skew-Symmetric

In summary, the conversation discussed the proof that the set of all 2x2 "symmetric" matrices is a subspace of Mat2x2(lR). It was mentioned that this proof could not be applied to skew-symmetric matrices, and an example was given to show this. The properties of skew-symmetric matrices and their operations were also discussed. The conversation ended with a question about where the proof may have gone wrong, and the suggestion to consider the kernel of any linear map as a subspace.
  • #1
mang733
5
0
I was able to show that the set of all 2x2 "symmetric" matrices is a subspace of Mat2x2(lR) using the 3 axioms. However, I wasn't able to do the same with skew-symmetric (A^T=-A). anyone can help? Thanks.
 
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  • #2
There's reason you can't! If A is the matrix with entries [0 1] and [-1 0] it is skew symmetric. What is A2?
 
  • #3
Subspace, Halls, squaring it doesn't matter.

0^t=0=-0

If A^t =-A, then (kA)^t = -kA, for k a scalar

(A+B)^t = A^t+B^t, so if they're skew symmetric then this is -A-B = -(A+B)

so where did you proof go wrong?
 
  • #4
the kernel of any linear map is a subspace.

consider the map taking A to (A + A^t). for symetric ones consider A goes to (A-A^t).
 
  • #5
thanks for the help guys.
 

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