Question

Problem 8 Suppose that the matrix equation Ax = b represents a consistent system of m equations in n unknowns and Xo is a specific solution of this system. Show that any solution of this system E can be written in the form x = xo + x1, where x1 is a solution of Ax = 0.

Answer 1

$Suppose,\overrightarrow{x}\,is \,a solution\,of\, the\,system\,A\overrightarrow{x}=\overrightarrow{b}$

$given,\overrightarrow{x}=\overrightarrow{x_{0}}+\overrightarrow{x_{1}}$

$so,A(\overrightarrow{x_{0}}+\overrightarrow{x_{1}})=\overrightarrow{b}$

$\Rightarrow A\overrightarrow{x_{0}}+A\overrightarrow{x_{1}}=\overrightarrow{b}$

$as,\overrightarrow{x_{0}}\,is \,a solution\,of\, the\,system\,A\overrightarrow{x}=\overrightarrow{b}$

$so,A\overrightarrow{x_{0}}=\overrightarrow{b}$

$now\,we\,can\,write,\overrightarrow{b}+A\overrightarrow{x_{1}}=\overrightarrow{b}$

$\Rightarrow A\overrightarrow{x_{1}}=\overrightarrow{0}$

$Therefore,\overrightarrow{x_{1}}\,is\,a\,solution\,of\,A\overrightarrow{x}=\overrightarrow{0}$

hence proved