Question

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 1

You're given that


`\mathbf{Ax}_1=\mathbf b`

and


`\mathbf{Ax}_0=\mathbf0`

Adding both systems together gives


`\mathbf{Ax}_1+\mathbf{Ax}_0=\mathbf b+\mathbf 0`

`\mathbf A(\mathbf x_1+\mathbf x_0)=\mathbf b`

which means
`\mathbf x_1+\mathbf x_0` is also a solution to
`\mathbf{Ax}=\mathbf b`.