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.