A question at the end of a chapter in a book I'm reading asked me to prove something which I'm thinking is a mistake because it seems to be impossible to prove.(adsbygoogle = window.adsbygoogle || []).push({});

'Prove that if two homogeneous systems of linear equations in two unknowns have the same solutions, then they are equivalent.'

The book defines equivalence to be 'two systems of linear equations are equivalent if each equation in each system is a linear combination of the equations in the other system'.

I can think of two systems which share the same solution (the trivial solution) that are not equivalent. That is the second system of homogeneous linear equations are not a combination of the first system of homogeneous linear equations but they both share the same solution.

system 1,

x - y = 0

2x + y = 0

system 2,

3x + y = 0

x + y = 0

Am I right in believing that their question is malformed or am I wrong in believing that you can not get x + y as a linear combination of c1(x - y) + c2(2x + y).

Thank you,

-= Dave

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# Impossible proof

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