Prove that if x1 and x2 are both solutions to a linear syste

Prove that if x1 and x2 are both solutions to a linear system Ax = b, then k(x1 - x2) is a solution to the homogenous linear system Ax = 0.

(where x1 means X sub 1)

Thank you!

Solution

if x1 and x2 are both solutions to a linear system Ax = b i.e.

we can write : Ax1 =b and Ax2 =b.

Further for homogeneous equation Ax=0, we can write:

A(kx1 - kx2) = kAx1 - kAx2 ----> k(b) - k(b) =0

So, A(k(x1 -x2) =0 .Hence k(x1 -x2 is a solution to the homogeneous linear system.