Linear independence and decompostion

[MotoPayton]

Explain why the method of decomposition when applied to the solution set of a homogeneous linear system always yields a linearly independent set of vectors whose span is the set of solutions....

Can someone explain this it seems reasonable but I can't seem to prove it to myself

[MotoPayton]

Maybe dependent spans do not exist because if that were to be the case the dependence would work itself out in the rref as a definite solution removing the dependence. So only independent spans form....

????

[Bacle]

Maybe if you explain to us what the method is, we can help you better, and, in doing the explaining, you may understand things better yourself.