General Linear Group: A belongs to Rn×n, the real n×n matrices,...
Which of these sets, if any, is actually a subspace of [FONT=MathJax_AMS]R[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Math]n[/FONT]?
I have tried assuming A & T to be a 2*2 Matrix, and verifiy that the matrices {TA. TAinv(T) etc} are invertible. I believe
None of these sets are actually subspaces of [FONT=MathJax_AMS]R[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Math]n[/FONT], unless [FONT=MathJax_Math]A[/FONT]is chosen to be the zero matrix. Notably, any subspace must contain the zero-element, which in this case is the matrix whose entries are all zero. It doesn't make sense to talk about the "subspaces of [FONT=MathJax_Math]G[/FONT][FONT=MathJax_Math]L[/FONT]", since [FONT=MathJax_Math]G[/FONT][FONT=MathJax_Math]L[/FONT] fails to be a vector space.
If [FONT=MathJax_Math]A belongs to [FONT=MathJax_AMS]R[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Math]n[/FONT], the real [FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Math]n[/FONT] matrices, and consider the sets [/FONT] [FONT=MathJax_Math]S[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]{[/FONT][FONT=MathJax_Math]T[/FONT][FONT=MathJax_Math]A[/FONT][FONT=MathJax_Main]∣[/FONT][FONT=MathJax_Math]T[/FONT][FONT=MathJax_Main]∈[/FONT][FONT=MathJax_Main]GL[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_AMS]R[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]}[/FONT][FONT=MathJax_Math] S[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]{[/FONT][FONT=MathJax_Math]T[/FONT][FONT=MathJax_Math]A[/FONT][FONT=MathJax_Math]T[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]∣[/FONT][FONT=MathJax_Math]T[/FONT][FONT=MathJax_Main]∈[/FONT][FONT=MathJax_Main]GL[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_AMS]R[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]}[/FONT][FONT=MathJax_Math] S[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]{[/FONT][FONT=MathJax_Math]T[/FONT][FONT=MathJax_Math]A[/FONT][FONT=MathJax_Main]∣[/FONT][FONT=MathJax_Math]T[/FONT][FONT=MathJax_Main]∈[/FONT][FONT=MathJax_Main]GL[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_AMS]Q[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]}[/FONT][FONT=MathJax_Math] S[/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]{[/FONT][FONT=MathJax_Math]T[/FONT][FONT=MathJax_Math]A[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]∣[/FONT][FONT=MathJax_Math]T[/FONT][FONT=MathJax_Main]∈[/FONT][FONT=MathJax_Main]GL[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_AMS]Q[/FONT][FONT=MathJax_Main]) [/FONT][FONT=MathJax_Math]S[/FONT][FONT=MathJax_Main]5[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]{[/FONT][FONT=MathJax_Math]A[/FONT][FONT=MathJax_Math]T[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]∣[/FONT][FONT=MathJax_Math]T[/FONT][FONT=MathJax_Main]∈[/FONT][FONT=MathJax_Main]GL[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_AMS]R[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]}[/FONT] |
I have tried assuming A & T to be a 2*2 Matrix, and verifiy that the matrices {TA. TAinv(T) etc} are invertible. I believe
None of these sets are actually subspaces of [FONT=MathJax_AMS]R[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main]×[/FONT][FONT=MathJax_Math]n[/FONT], unless [FONT=MathJax_Math]A[/FONT]is chosen to be the zero matrix. Notably, any subspace must contain the zero-element, which in this case is the matrix whose entries are all zero. It doesn't make sense to talk about the "subspaces of [FONT=MathJax_Math]G[/FONT][FONT=MathJax_Math]L[/FONT]", since [FONT=MathJax_Math]G[/FONT][FONT=MathJax_Math]L[/FONT] fails to be a vector space.
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