Question

suppose p and q are solutions to Ax=b, prove that p-q is in the null space of A

Answer 1

Consider a linear transformation Ax=0, then the solution set for x is called the null space of A.
Ax=b
as p and q are the solutions to Ax=b
so we have Ap=b --> (1) and Aq=b --> (2)
next we have (1) - (2) and we get Ap-Aq = b-b
so we get A(p-q) = 0. compare it to Ax=0 so we can say that p-q is the null space of A.

Answer 2

This makes so much sense now ! thank you so much !