I've just begun to work learn Linear Algebra on my own through Hoffman and Kunze's book and the first problem set already has a question that I can't solve:
Prove that if two homogeneous systems of linear equations in two unknowns have the same solutions, then they are equivalent.
I can't seem to figure out how to prove this without resorting to case work where you account for the cases where one of the coefficients are zero and when both are.
Is there an elegant way to prove in general that when two systems of linear equations have the same solutions, they are equivalent? The converse is obvious enough though.
Definition of equivalence from the text :
Let us say that two systems of linear equations are equivalent if each equation in each system is a linear combination of the equations in the other system.