Let A be an m times n matrix and let b elementof Rm Show tha

Let A be an m times n matrix, and let b elementof R^m. Show that if u and v elementof R^n are both solutions of Ax = b, then u - v is a solutions of A_x = 0. Suppose u is a solutions of A_x = 0 and p is a solution of A_x = b. Show that u + p is a solution of A_= b.

Solution

a,

u and v are solutions to Ax=b

HEnce, Au=b and Av=b

Au-Av=b-b=0

Au-Av=A(u-v)=0

HEnce, u-v is solution to Ax=0

b.

Au=0

and

Ap=b

A(u+p)=Au+Ap=0+b=b

HEnce, u+p is a solutin to Ax=b