shree012
Answered

let Ax=b be any consistent system of linear equations, and let x1 be a fixed solution. show that every solution to the system can be written in form x=x1+x2, where x0 is a solution to Ax=0. show that every matrix of this form is a solution.
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Answer :

LammettHash
You're given that

[tex]\mathbf{Ax}_1=\mathbf b[/tex]

and

[tex]\mathbf{Ax}_0=\mathbf0[/tex]

Adding both systems together gives

[tex]\mathbf{Ax}_1+\mathbf{Ax}_0=\mathbf b+\mathbf 0[/tex]
[tex]\mathbf A(\mathbf x_1+\mathbf x_0)=\mathbf b[/tex]

which means [tex]\mathbf x_1+\mathbf x_0[/tex] is also a solution to [tex]\mathbf{Ax}=\mathbf b[/tex].

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