Echleon Matrices : Linear Algebra

[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.

[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.

[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?

[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.

[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.

[lime]

Yes since their reduced echelon form matrices must coincide.

[mobius70]

Thanks for input, lime.