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Echleon Matrices : Linear Algebra

Posted: Sat Sep 06, 2008 2:20 pm
by mobius70
Straight to the problem. :)

Suppose R and R* are 2x3 row-reduced 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.

Posted: Sat Sep 06, 2008 3:26 pm
by mobius70
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.

Posted: Sat Sep 06, 2008 3:59 pm
by Nameless
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?

Posted: Sat Sep 06, 2008 8:42 pm
by lime
Suppose R and R* are 2x3 row-reduced 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.

Posted: Mon Sep 08, 2008 12:48 am
by mobius70
Hi Lime

I understand that "two equivalent systems (homogeneous or non-homogeneous) have the same solution sets".

The problem is if the systems have same solutions are they equivalent.

Plz explain more on this.

Posted: Tue Sep 09, 2008 3:21 am
by lime
Yes since their reduced echelon form matrices must coincide.

Posted: Tue Sep 09, 2008 4:37 am
by mobius70
Thanks for input, lime.