【考研数学一·线性代数(4)】线性方程组

1.线性方程组理论

1.1.齐次线性方程组

$$\{\begin{array}{l}{a}_{11}{x}_1+a_{12}{x}_2+\cdots +a_{1n}{x}_n=0\\ a_{21}{x}_1+a_{22}{x}_2+\cdots +a_{2n}{x}_n=0\\ \cdots \\ a_{m1}{x}_1+a_{m2}{x}_2+\cdots +a_{mn}{x}_n=0\end{array}$$

$$\begin{gathered} 向量形式:x_1{\alpha }_1+x_2{\alpha }_2+\cdots +x_n{\alpha }_n=0,{\alpha }_j=[a_{1j},a_{2j},\cdots ,a_{mj}{]}^T(j=1,2,\cdots ,n) \\ 矩阵形式:A_{m\times n}x=0,A_{m\times n}=\left[\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& & ⋮\\ a_{m1}& a_{m2}& \cdots & a_{mn}\end{array}\right],x=[x_1,x_2,\cdots ,x_n{]}^T \end{gathered}$$

1. 解的性质

   若${\xi }_1,{\xi }_2$是方程组$Ax=0$的解，则$x=k_1{\xi }_1+k_2{\xi }_2$也是它的解($k_1,k_2\in R$).

2. 基础解系

   ${\xi }_1,{\xi }_2,\cdots ,{\xi }_s$是方程组$Ax=0$的一个基础解系$\{\begin{array}{l}x{i}_1,{\xi }_2,\cdots ,{\xi }_s是Ax=0的一组解向量\\ {\xi }_1,{\xi }_2,\cdots ,{\xi }_s线性无关\\ s=n-r(A)\end{array}$

3. 有解的条件

   • $r(A)=n$(列满秩)$\iff$只有零解
   • $r(A)<n\iff$有非零解，通解$x=k_1{\xi }_1+k_2{\xi }_2+\cdots +k_{n-r}{\xi }_{n-r}(r=r(A))$

1.2.非齐次线性方程组

{ a 11 x 1 + a 12 x 2 + ⋯ + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + ⋯ + a 2 n x n = b 2 ⋯ a m 1 x 1 + a m 2 x 2 + ⋯ + a m n