System of linear equations
Definition
\[ \begin{aligned} a_{11} x_{1}+a_{12} x_{2}+\cdots+a_{1 n} x_{n} &=b_{1} \\ a_{21} x_{1}+a_{22} x_{2}+\cdots+a_{2 n} x_{n} &=b_{2} \\ \vdots \\ a_{m 1} x_{1}+a_{m 2} x_{2}+\cdots+a_{m n} x_{n} &=b_{m} \end{aligned} \]
\[ A=\left[\begin{array}{cccc}{a_{11}} & {a_{12}} & {\cdots} & {a_{1 n}} \\ {a_{21}} & {a_{22}} & {\cdots} & {a_{2 n}} \\ {\vdots} & {\vdots} & {\vdots} & {\vdots} \\ {a_{m 1}} & {a_{m 2}} & {\cdots} & {a_{m n}}\end{array}\right], \vec{b}=\left[b_{1} b_{2} \cdots b_{m}\right], \vec{x}=\left[\begin{array}{ccc}{x_{1}} & {x_{2}} & {\cdots} & {x_{n}}\end{array}\right] \]
\(A:\)Coefficient Matrix(系数矩阵)，\(\vec b\): Constants(常数)， \(\vec x\): Variables
Example
\[ .80x_1 + .60x_2 + .40x_3 = 5\\ .20x_1 + .40x_2 + .60x_3 = 3 \]
Solution Set
Definition
The set of all solutions of a system of linear equations is called Solution Set
\[ solution~set = \lbrace \left[\begin{array}{cc}x_1\\x_2\\\vdots\\ x_n \end{array}\right]\in \mathcal{R}^n|x_1, x_2, \cdots, x_n~satisfy~the~system~of~linear~equations \rbrace \]
Example
A linear equation in two variables x and y has the form \(ax + by = c\), when at least one of a and b is nonzero, this is the equation of a line in the \(xy\)-plane, Thus a system of 2 linear equations in the variables x and y consists of a pair of equations, each of describe a line in the plane.(两个二元一次方程的解可以理解为二维平面上两条直线的交点情况)
\[ a_1 x + b_1 y = c_1 \qquad \text{is the equation of line $\mathcal{L_1}$} \\ a_2 x + b_2 y = c_2 \qquad \text{is the equation of line $\mathcal{L_2}$} \]
only one solution （两直线相交）
infinitely many solutions （两直线重合）
no solution （两直线平行不重合）
Statement
Every system of linear equations has no solution, exactly one solution, or infinitely many solutions.
A system of linear equations that has one or more solutions is called consistent; Otherwise, the system is inconsistent.
Two system of linear equations are called equivalent if they have exactly the same solution set.
Elementary Row Operations
Definition
Interchage any two rows of the matrix. (Interchange operation)
Multiply every entry of some row of the matrix by the same nonzero scalar. (scaling operation)
Add a multiple of one row of the matrix to another row. (row addition operation)
\(\mathbf{A}x = b\)，$ [\mathbf{A}~b]$ is called the augmented matrix (增广矩阵)
Property
Every elementary row operations are reversble (基本行操作都是可逆的)
\[ \mathbf{A}x = b \Longleftrightarrow \mathbf{A'}x = b' \quad \text{(equivalent)}\\ [\mathbf{A}~b] \longleftrightarrow [\mathbf{A'~}b'] \quad \text{(elmentary row opeations)} \]
Elementary row operations taken on the system of linear equations as well as on the corresponding augmented matrices. （在线性方程组上的基本行操作等价于在增广矩阵上进行相关的操作）
Row echelon form
Definition
- Each nonzero row lies above every zero row. （非零行一定在所有零行的上面）
- The leading entry of a nonzero row lies in a column to the right of the column containing the leading entry of any preceding row.
- If a column contains the leading entry of some row, the all entries of that column below the leading entry are 0. （如果某一列包含某行的首项，那么该首项下面的所有元素都为0）
if a matrix also satisfied the following two additional conditions, we say that it is in reduced echelon form.
- If a column contains the leading entry of some row, the all the others entries of that column are 0.
（如果某一列包含某行的首项，那么该列除此首项外的所有元素都是0） - The leading entry of each nonzero row is 1. （所有非零行的首项都是1）
Example
考虑行的leading entry，可以将每行的leading entry成为basic variables, 改行其他的entry就可以称为free variables, free variables可以任意取值，但是不一定在每个system of linear equations里面都存在free variable，考虑这样的方程组
\[ \begin{aligned} x1 - 3x_2 ~~~~~~~+ 2x_4 ~~~~~~ = 7 \\ x_3 + 6x_4 ~~~~~~ = 9\\ x_5 = 2\\ 0 = 0\\ \end{aligned} \]
The resulting equations
\[ \begin{aligned} &x_1 = 3x_2 - 2x_4 + 7 \\ &x_2 =\text{free variable}\\ &x_3 = 9 - 6x_4 \\ &x_4 = \text{free variable} \\ &x_5 = 2 \end{aligned} \]
可以得到方程组的通解(general solution)
\[ \begin{bmatrix} x_1 \\ x_2\\x_3\\x_4\\x_5\end{bmatrix} = \begin{bmatrix} 7 \\ 0 \\ 9 \\ 0 \\ 2\end{bmatrix} + x_2\begin{bmatrix} 3\\1\\0\\0\\0\end{bmatrix} + x_4\begin{bmatrix} -2 \\ 0 \\ -6 \\ 1 \\ 0\end{bmatrix} \]
Summary
求解线性方程组的一般步骤就是将增广矩阵(augmented matrix)其化为最简的行阶梯型\([\mathbf{A}~b]\xrightarrow{}[\mathbf{R}~c]\)
然后通过basic variable和free variable来求通解