Definition
Linear Equation 线性方程
所有变量都是**一次方**，并且**与常量相乘**
$$a_{1}x_{1}+a_{2}x_{2}+\dots+a_{n}x_{n}=b$$
System of Linear Equation 线性方程组
几个包含相同变量的Linear Equation组成的式子。
Forms
Linear Equation->Vector Form->Matrix Form->Augmented Matrix Form
$$ \begin{cases}3x_{1}+x_{2}=4\\6x_{1}+5x_{2}=2\end{cases} \to \begin{pmatrix}3\\6\end{pmatrix}x_{1}+ \begin{pmatrix}1\\5\end{pmatrix}x_{2}= \begin{pmatrix}4 \\2\end{pmatrix} \to \begin{bmatrix}3&1\\6&5\end{bmatrix} \begin{pmatrix}x_{1}\\x_{2}\end{pmatrix}= \begin{pmatrix}4\\2\end{pmatrix} \to \left[\begin{array}{cc|c}3 & 1 & 4 \\6 & 5 & 2\end{array}\right] $$Solutions
Consistent vs inconsistent - consistent: has solution - one solution - infinitely many solutions - inconsistent: no solution
Homogeneous vs Non-homogeneous - Homogeneous - $A\overrightarrow{x}=0$ - at least one solution, where $\overrightarrow{x}=0$ (trivial solution) - non trivial solution exits only if theres at least one free variable - Non-homogeneous - $A\overrightarrow{x}=\overrightarrow{b}$
Solve
Gaussian Elimination with row operation
Echelon Form& Row Reduction
Row Reduction
把矩阵变成Reduced Echelon Form
1. 从左往右，把pivot都变成1，下面都变成0
2. 从右往左，把pivot上方的数字都变成0