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padge
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Joined: 15 Oct 2006
Posts: 13
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 Posted: Mon, 13 Nov 2006 06:05:14 UTC Post subject: Vector Spaces Q's |
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Here are some questions I'm working on and I'm having one heck of a time trying to start them. I need a lot more practice on this stuff... help would be much appreciated!
1)
Which of the following sets are linear subspaces of the space M of n x n matrices.
a) 
b) 
c) 
2)
Let V and W be (real) vector spaces and f : V -> W be a linear map.
a) Show that the range of f (i.e. f(V)) is a linear subspace of W.
b) Show that the kernel of f (i.e.  ) is a linear subspace of V.
3)
For integers 0 < k < 3 we consider set 3 x 3 matrices A which satisfy  .
a) Show that this set of matrices is no subspace of the 3x3 matrices.
b) Show that the only possible eigenvalues of such an A are 0 and 1.
c) Let f :  -> be defined by f(x) = Ax. Show that each vector of the range of f is orthogonal to all vectors of the kernel of f. |
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Matt
Member of the 'S.O.S. Math' Hall of Fame
Joined: 01 Oct 2003
Posts: 8507
Location: Sacramento, CA
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| Posted: Mon, 13 Nov 2006 07:22:22 UTC Post subject: Re: Vector Spaces Q's |
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| padge wrote: | 1)
Which of the following sets are linear subspaces of the space M of n x n matrices.
a) |
Not a subspace. Find matrices A and B such that det(A)=0 and det(B)=0 but det(A+B) is not zero.
| padge wrote: | | b) |
(1) It is clear that the zero matrix belongs to the set.
(2) Let A and B be matrices such that tr(A)=0 and tr(B)=0. Then tr(A+B)=tr(A)+tr(B)=0. Thus A+B belongs to the set.
(3) Finally, let c be a real number and A be a matrix for which tr(A)=0. Then tr(cA)=c*tr(A)=0, so that cA belongs to the set as well.
With these three conditions satisfied, it follows that this is a subspace.
| padge wrote: | | c) |
(1) It is clear that the transpose of the zero matrix is the same as the negative of the zero matrix.
(2) Let A and B be matrices such that  and  Then
so that A+B belongs to the set as well.
(3) Finally let c be a real number and A be a matrix such that  Then
so that cA belongs to the set as well.
With these three conditions satisfied, it follows that this is a subspace. |
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Matt
Member of the 'S.O.S. Math' Hall of Fame
Joined: 01 Oct 2003
Posts: 8507
Location: Sacramento, CA
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| Posted: Mon, 13 Nov 2006 07:35:00 UTC Post subject: |
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| padge wrote: | 2)
Let V and W be (real) vector spaces and f : V -> W be a linear map.
a) Show that the range of f (i.e. f(V)) is a linear subspace of W. |
(1) 0 belongs to f(V) since f(0)=0.
(2) Let  Then there are  such that f(a)=x and f(b)=y. Hence:
so that  as well.
(3) Finally, let c be a real number and  Then there exists an  such that f(a)=x. Hence:
so that  as well.
With these three conditions satisfied, it follows that f(V) is a subspace.
| padge wrote: | | b) Show that the kernel of f (i.e. ) is a linear subspace of V. |
(1) 0 belongs to  since 
(2) Let  Then  and  Hence  so that  as well.
(3) Finally, let c be a real number and  Then  Hence  so that  as well.
With these three conditions satisfied, it follows that is a subspace.
Last edited by Matt on Mon, 13 Nov 2006 10:14:32 UTC; edited 1 time in total |
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padge
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Joined: 15 Oct 2006
Posts: 13
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| Posted: Mon, 13 Nov 2006 09:22:53 UTC Post subject: |
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Thank you Matt your help is greatly appreciated. I think I would do best to hire a tutor if i can find one.
Do you have anything for 3? |
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Kungsman
Member of the 'S.O.S. Math' Hall of Fame
Joined: 04 Sep 2004
Posts: 3401
Location: Uppsala, Sweden
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| Posted: Mon, 13 Nov 2006 10:13:24 UTC Post subject: |
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| padge wrote: | | Do you have anything for 3? |
I'm sure he has, but maybe he didn't want to spoil all the fun for you. Have you tried something yourself, especially after having seen Matthew's responses?
(a) The set fails to be a linear subspace for more than one reason. Can you find two matrices, both equal to their squares and transposes and of equal trace, whose sum fails to equal its square (and whose sum will have a different trace). Hint: Think diagonal matrices.
(b) Let  be an eigenvalue coresponding to the eigenvector v. We know  . What can you say about the left hand side? Conclusion?
(c) I assume you use the standard inner product. If v=Ax and Aw=0 then
(I let you finish.) |
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