 |
|
| Science Forum Index » Mathematics Forum » Lin Alg and Direct Sums |
|
Page 1 of 1 |
|
| Author |
Message |
| Mike Lukas |
 Posted: Sat Feb 12, 2005 4:40 pm |
|
|
|
Guest
|
Hello,
I am working on and am stuck on a problem which goes: We have an m x n
matrix A over F. Prove that F^n is the direct sum of the null space and the
row space of A. Please no full solutions, I just need a few hints to get me
going in the right direction.
Thanks,
Mike L |
|
|
| Back to top |
|
|
|
| Robin Chapman |
| Posted: Sun Feb 13, 2005 4:15 am |
|
|
|
Guest
|
Mike Lukas wrote:
[quote:b4938aee62]Hello,
I am working on and am stuck on a problem which goes: We have an m x n
matrix A over F. Prove that F^n is the direct sum of the null space and
the
row space of A.
[/quote:b4938aee62]
Hmmm. The row space is a subspace of F^n considered as row vectors.
The null space could refer to the set of vectors v with vA = 0,
this is a subspace of F^m considered as row vectors. Or it
could refer to the set of vectors w with wA = 0, this is a subspace
of F^n considered as column vectors. In any case the null space does
not lie in the same space as the row space, so I don't understand
why you say F^n *is* the sum of both. What is true is that
F^n is isomorphic to the (external) direct sum of the row and null spaces.
This is a dimension count.
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
"Elegance is an algorithm"
Iain M. Banks, _The Algebraist_ |
|
|
| Back to top |
|
|
|
| Guest |
| Posted: Sun Feb 13, 2005 4:15 am |
|
|
|
|
In article <cun5qq$8fh$3@news6.svr.pol.co.uk>,
Robin Chapman <rjc@ivorynospamtower.freeserve.co.uk> writes:
[quote:857b6de5b3]Mike Lukas wrote:
Hello,
I am working on and am stuck on a problem which goes: We have an m x n
matrix A over F. Prove that F^n is the direct sum of the null space and
the
row space of A.
Hmmm. The row space is a subspace of F^n considered as row vectors.
The null space could refer to the set of vectors v with vA = 0,
this is a subspace of F^m considered as row vectors. Or it
could refer to the set of vectors w with wA = 0, this is a subspace
of F^n considered as column vectors. In any case the null space does
not lie in the same space as the row space, so I don't understand
why you say F^n *is* the sum of both. What is true is that
F^n is isomorphic to the (external) direct sum of the row and null spaces.
This is a dimension count.
[/quote:857b6de5b3]
But you could identify the column- and row-spaces F^n, and thereby
consider both the row space and the (right) nullspace as subspaces
of F^n. Then his result is just saying that a subspace of F^n
is disjoint from its orthogonal complement using the standard inner
product.
But hang on, what is F? If F is the complex numbers, then we can choose
A = ( 1 i), and the nullspace is spanned by (1 i)^T.
Shomething wrong, surely?
Derek Holt. |
|
|
| Back to top |
|
|
|
|
|
All times are GMT - 5 Hours
The time now is Sun Dec 06, 2009 10:59 pm
|
|