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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.

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    What's the book's definition of "equivalent"? Usually it simply means having the same set of solutions...2011-06-18
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    @Lundmark: here it is probably an equivalence relation on the matrices associated with the equations, defined by some row/column operations. So the thing to prove is that those operations don't change the solution space. Is this what you had in mind, @Herman?2011-06-18
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    @Marek: Well, the way Hoffman and Kunze define equivalent is that each equation in one system is a linear combination of equations in the other.2011-06-18

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