# 非齐次线性方程组不同解向量的系数相加等于 1 时，相加所得的向量也是该方程的解

## 二、解析

\textcolor{yellow}{ \begin{aligned} \boldsymbol{A} \boldsymbol{\eta}_{1} & = b \\ \boldsymbol{A} \boldsymbol{\eta}_{2} & = b \\ \boldsymbol{A} \boldsymbol{\eta}_{3} & = b \end{aligned} }

\textcolor{yellow}{ \begin{aligned} \boldsymbol{A} (\textcolor{magenta}{k_{1}} \boldsymbol{\eta}_{1} ) & = \textcolor{magenta}{k_{1}} b \\ \boldsymbol{A} (\textcolor{magenta}{k_{2}} \boldsymbol{\eta}_{2} ) & = \textcolor{magenta}{k_{2}} b \\ \boldsymbol{A} (\textcolor{magenta}{k_{3}} \boldsymbol{\eta}_{3} ) & = \textcolor{magenta}{k_{3}} b \end{aligned} }

\textcolor{yellow}{ \begin{aligned} \boldsymbol{A} ( & \textcolor{magenta}{k_{1}} \boldsymbol{\eta}_{1} + \textcolor{magenta}{k_{2}} \boldsymbol{\eta}_{2} + \textcolor{magenta}{k_{3}} \boldsymbol{\eta}_{3}) \\ = & \textcolor{magenta}{k_{1}} b + \textcolor{magenta}{k_{2}} b + \textcolor{magenta}{k_{3}} b \\ = & (\textcolor{magenta}{k_{1} + k_{2} + k_{3}}) b \end{aligned} }

$$\textcolor{magenta}{k_{1} + k_{2} + k_{3}} = \textcolor{springgreen}{1}$$

\textcolor{yellow}{ \begin{aligned} \boldsymbol{A} (\textcolor{magenta}{k_{1}} \boldsymbol{\eta}_{1} + \textcolor{magenta}{k_{2}} \boldsymbol{\eta}_{2} + \textcolor{magenta}{k_{3}} \boldsymbol{\eta}_{3}) & = \textcolor{springgreen}{1} \times b \\ & = b \end{aligned} }