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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.

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Answer:

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

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

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