Forum for the GRE subject test in mathematics.

mobius70
 Posts: 19
 Joined: Thu Jan 24, 2008 1:11 am
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by mobius70 » Sat Sep 06, 2008 2:20 pm
Straight to the problem.
Suppose R and R* are 2x3 rowreduced echleon matrices and that the systems RX = 0 and R*X = 0 have exactly the same solutions. Prove that R = R*.
I understand that one way of going about it can be making cases for 2x3 echleon matrices and then showing no two different echleon froms have same solution set. But is there some more elegant way.
Thanks in advance for solutions / approach.
P.S : Source is Hoffman and Kunze.

mobius70
 Posts: 19
 Joined: Thu Jan 24, 2008 1:11 am
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by mobius70 » Sat Sep 06, 2008 3:26 pm
Another related question .. which might be of importance here ..
Is the following statement true?
Two homogeneous systems are equivalent if and only if they have exactly the same solutions.

Nameless
 Posts: 128
 Joined: Sun Aug 31, 2008 4:42 pm
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by Nameless » Sat Sep 06, 2008 3:59 pm
Another related question .. which might be of importance here ..
Is the following statement true?
Two homogeneous systems are equivalent if and only if they have exactly the same solutions.
I don't know what is the definition of equivalent homogeneous system ? Can you give me the definition?

lime
 Posts: 129
 Joined: Tue Dec 04, 2007 2:11 am
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by lime » Sat Sep 06, 2008 8:42 pm
Suppose R and R* are 2x3 rowreduced echleon matrices and that the systems RX = 0 and R*X = 0 have exactly the same solutions. Prove that R = R*.
To avoid ambiguity you would better use "~" for equivalence of matrices.
Since
AX=0 and BX=0
matrices A and B have the same null space
hence they have the same row space
hence they are equivalent:
A~B.
Is the following statement true?
Two homogeneous systems are equivalent if and only if they have exactly the same solutions.
Yes. It is actually another definition of equivalence of systems of linear equations. Moreover, it still holds if systems are non homogeneous.

mobius70
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 Joined: Thu Jan 24, 2008 1:11 am
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by mobius70 » Mon Sep 08, 2008 12:48 am
Hi Lime
I understand that "two equivalent systems (homogeneous or nonhomogeneous) have the same solution sets".
The problem is if the systems have same solutions are they equivalent.
Plz explain more on this.

lime
 Posts: 129
 Joined: Tue Dec 04, 2007 2:11 am
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by lime » Tue Sep 09, 2008 3:21 am
Yes since their reduced echelon form matrices must coincide.

mobius70
 Posts: 19
 Joined: Thu Jan 24, 2008 1:11 am
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by mobius70 » Tue Sep 09, 2008 4:37 am
Thanks for input, lime.