Definition
[!NOTE] Linear Equation 线性方程
所有变量都是一次方，并且与常量相乘
$$a_{1}x_{1}+a_{2}x_{2}+\dots+a_{n}x_{n}=b$$
[!NOTE] 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
[!NOTE] 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