线性方程组

1.齐次线性方程组

(1)、一般形式:
\[\left\{\begin{matrix}a_{11}x_{1}&+a_{12}x_{2}&...&+a_{1n}x_{n}=0\\ a_{21}x_{2}&+a_{22}x_{2}&...&+a_{2n}x_{n}=0\\ ...\\a_{m1}x_{m}&+a_{m2}x_{2}&...&+a_{mn}x_{n}=0\\\end{matrix}\right.\]

(2)、矩阵形式: 
\(A_{m\times n}x=b\),其中系数矩阵和向量x分别为
\[A=\begin{bmatrix}
a_{11}&+a_{12}& ...&+a_{1n}\\ 
a_{21}&+a_{22}& ...&+a_{2n}\\ 
...\\
a_{m1}&+a_{m2}& ...&+a_{mn}\\
\end{bmatrix}  x=\begin{bmatrix}x_{1}\\ x_{2}\\ ...\\x_{n}\\\end{bmatrix}\]
(3)、向量形式: 
将矩阵每一列看成一个向量\(\alpha\),则\(A=\begin{bmatrix}
\alpha_{1}& \alpha_{2}&...&\alpha_{n}\\ 
\end{bmatrix}\),所以
\[\begin{bmatrix}
\alpha_{1}& \alpha_{2}&...&\alpha_{n}\\ 
\end{bmatrix}\times \begin{bmatrix}
x_{1}\\ 
x_{2}\\ 
...\\
x_{n}\\
\end{bmatrix}=x_{1}\alpha_{1}+x_{2}\alpha_{2}+...+x_{n}\alpha_{n}=0\]

2.非齐次线性方程组
(1)、一般形式:

 \[\left\{\begin{matrix}
a_{11}x_{1}&+a_{12}x_{2}& ...&+a_{1n}x_{n} = b_{1}\\ 
a_{21}x_{2}&+a_{22}x_{2}& ...&+a_{2n}x_{n} = b_{2}\\ 
...\\
a_{m1}x_{m}&+a_{m2}x_{2}& ...&+a_{mn}x_{n} = b_{m}\\
\end{matrix}\right.\]
(2)、矩阵形式: 

\(A_{m\times n}x=b\),其中系数矩阵和向量x分别为
\[A=\begin{bmatrix}
a_{11}&+a_{12}& ...&+a_{1n}\\ 
a_{21}&+a_{22}& ...&+a_{2n}\\ 
...\\
a_{m1}&+a_{m2}& ...&+a_{mn}\\
\end{bmatrix}    x=\begin{bmatrix}x_{1}\\ x_{2}\\ ...\\x_{n}\\\end{bmatrix}\]
(3)、向量形式: 
将矩阵每一列看成一个向量\(\alpha\),则\(A=\begin{bmatrix}
\alpha_{1}& \alpha_{2}&...&\alpha_{n}\\ 
\end{bmatrix}\),所以
\[\begin{bmatrix}
\alpha_{1}& \alpha_{2}&...&\alpha_{n}\\ 
\end{bmatrix}\times \begin{bmatrix}
x_{1}\\ 
x_{2}\\ 
...\\
x_{n}\\
\end{bmatrix}=x_{1}\alpha_{1}+x_{2}\alpha_{2}+...+x_{n}\alpha_{n}=b\]