Question

Consider a matrixA∈ Mm,nand a vectorb∈Rm, and letAx=bbe the correspondingsystem of linear equations. The systemAx=0is then called thehomogeneous systemassociated toAx=b. LetSbe the set of solutions ofAx=b, and letS0be the setof solutions of the associated homogeneous system. Prove that ifx∈ Sandu∈ S0thenx+u∈S.

Answer 1

Remember that if u is a solution of the linear system Ax=b then satisfies that Au=b. Then, since
`x\in S`,
`Ax=b`, and since
`u\in S_0` then
`Au=0`.

Now we need show that A(x+u)=b. Observe that
`A(x+u)=Ax+Au=b+0=b`, this implies that
`x+u\in S`