线性方程组 Linear Equations
简介 Introduction
在线性代数中，两个重要的方程组形式是 和 。这两个方程组涉及到矩阵 和向量 、，它们的解集有着不同的性质和几何意义。理解这些解集对于线性代数的学习和应用至关重要。
In linear algebra, two important forms of systems of equations are and . These systems involve a matrix and vectors and , and their solution sets have different properties and geometric meanings. Understanding these solution sets is crucial for learning and applying linear algebra.
齐次线性方程组 Homogeneous System
定义与性质 Definition and Properties
-
定义 Definition：
A = \begin{pmatrix}
1 & 2 & 1 \
2 & 4 & 2 \
3 & 6 & 3
\end{pmatrix} \rightarrow \text{RREF} \rightarrow \begin{pmatrix}
1 & 2 & 1 \
0 & 0 & 0 \
0 & 0 & 0
\end{pmatrix}
通过 RREF，可以发现 $x_1 + 2x_2 + x_3 = 0$，即 $$x = t_1 \begin{pmatrix} -2 \\ 1 \\ 0 \end{pmatrix} + t_2 \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}$，其中 $t_1, t_2 \in \mathbb{R}$$。 By RREF, we find that $x_1 + 2x_2 + x_3 = 0$, hence $$x = t_1 \begin{pmatrix} -2 \\ 1 \\ 0 \end{pmatrix} + t_2 \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}$, where $t_1, t_2 \in \mathbb{R}$$. 2. **使用分块矩阵和转置矩阵 Constructing Solution with Block Matrix and Transpose Matrix**： 假设矩阵 $A$ 的 RREF 可以表示为分块矩阵：\text{RREF}(A) = \begin{pmatrix}
I & B \
O & O
\end{pmatrix}
其中 $I$ 是单位矩阵，$B$ 是适当大小的矩阵，$O$ 是零矩阵。用 $B$ 的转置 $B^T$ 和单位矩阵来构造零空间的基础解系。 Suppose the RREF of matrix $A$ can be represented as a block matrix:\text{RREF}(A) = \begin{pmatrix}
I & B \
O & O
\end{pmatrix}
where $I$ is the identity matrix, $B$ is an appropriately sized matrix, and $O`$ is the zero matrix. Use the transpose of $B$, $B^T$, and the identity matrix to construct the basis for the null space. 例 Example：A = \begin{pmatrix}
1 & 2 & 1 & 3 \
2 & 4 & 2 & 6 \
3 & 6 & 3 & 9
\end{pmatrix} \rightarrow \text{RREF} \rightarrow \begin{pmatrix}
1 & 2 & 1 & 3 \
0 & 0 & 0 & 0 \
0 & 0 & 0 & 0
\end{pmatrix}
从 RREF 可以看到 $x_2, x_3, x_4$ 是自由变量。构造 $B^T$：B = \begin{pmatrix}
2 & 1 & 3
\end{pmatrix}, \quad B^T = \begin{pmatrix}
2 \
1 \
3
\end{pmatrix}
x = t_1 \begin{pmatrix}
-2 \
1 \
0 \
0
\end{pmatrix} + t_2 \begin{pmatrix}
-1 \
0 \
1 \
0
\end{pmatrix} + t_3 \begin{pmatrix}
-3 \
0 \
0 \
1
\end{pmatrix}
其中 $t_1, t_2, t_3 \in \mathbb{R}$。 From the RREF, we see that $x_2, x_3, x_4$ are free variables. Construct $B^T$:B = \begin{pmatrix}
2 & 1 & 3
\end{pmatrix}, \quad B^T = \begin{pmatrix}
2 \
1 \
3
\end{pmatrix}
x = t_1 \begin{pmatrix}
-2 \
1 \
0 \
0
\end{pmatrix} + t_2 \begin{pmatrix}
-1 \
0 \
1 \
0
\end{pmatrix} + t_3 \begin{pmatrix}
-3 \
0 \
0 \
1
\end{pmatrix}
where $t_1, t_2, t_3 \in \mathbb{R}$. ## 非齐次线性方程组 $Ax = b$ Non-Homogeneous System $Ax = b$ ### 定义与性质 Definition and Properties 1. **定义 Definition**：Ax = b
这里，$b$ 是一个 $m$ 维列向量。 Here, $b$ is an $m$-dimensional column vector. 2. **解集 Solution Set**： 如果 $Ax = b$ 有解，那么解集可以表示为一个特解 $x_p$ 加上齐次方程组 $Ax = 0$ 的解空间，即x = x_p + N(A)
If $Ax = b$ has a solution, the solution set can be expressed as a particular solution $x_p$ plus the solution space of the homogeneous system $Ax = 0$, i.e.,x = x_p + N(A)
### 求解方法 Solution Methods 1. **行简化 Row Reduction**： 将增广矩阵 $[A | b]$ 进行行简化，直到得到简化行阶梯形矩阵。 Perform row reduction on the augmented matrix $[A | b]$ until you obtain the reduced row echelon form (RREF). 例 Example：A = \begin{pmatrix}
1 & 2 & 1 \
2 & 4 & 2 \
3 & 6 & 3
\end{pmatrix}, \quad b = \begin{pmatrix}
1 \
2 \
3
\end{pmatrix} \rightarrow [A | b] = \begin{pmatrix}
1 & 2 & 1 & 1 \
2 & 4 & 2 & 2 \
3 & 6 & 3 & 3
\end{pmatrix} \rightarrow \text{RREF} \rightarrow \begin{pmatrix}
1 & 2 & 1 & 1 \
0 & 0 & 0 & 0 \
0 & 0 & 0 & 0
\end{pmatrix}
通过 RREF，可以发现 $x_1 + 2x_2 + x_3 = 1$，解集为 $$x = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} + t_1 \begin{pmatrix} -2 \\ 1 \\ 0 \end{pmatrix} + t_2 \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}$$，其中 $t_1, t_2 \in \mathbb{R}$。 By RREF, we find that $x_1 + 2x_2 + x_3 = 1$, hence the solution set is $$x = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} + t_1 \begin{pmatrix} -2 \\ 1 \\ 0 \end{pmatrix} + t_2 \begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}$$, where $t_1, t_2 \in \mathbb{R}$。 ## Rouché-Capelli Theorem ### Definition / 定义 **Rouché-Capelli Theorem**: This theorem states that a system of linear equations $\mathbf{A}\mathbf{x} = \mathbf{b}$ has at least one solution if and only if the rank of the coefficient matrix $\mathbf{A}$ is equal to the rank of the augmented matrix $[\mathbf{A} | \mathbf{b}]$.\text{rank}(\mathbf{A}) = \text{rank}([\mathbf{A} | \mathbf{b}])
**Rouché-Capelli 定理**：该定理指出，对于线性方程组 $\mathbf{A}\mathbf{x} = \mathbf{b}$，当且仅当系数矩阵 $\mathbf{A}$ 的秩等于增广矩阵 $[\mathbf{A} | \mathbf{b}]$ 的秩时，方程组至少有一个解。\text{rank}(\mathbf{A}) = \text{rank}([\mathbf{A} | \mathbf{b}])
### Properties / 性质 1. **Consistent System**: If $\text{rank}(\mathbf{A}) = \text{rank}([\mathbf{A} | \mathbf{b}])$, the system is consistent (i.e., it has at least one solution). - **一致性系统**：如果 $\text{rank}(\mathbf{A}) = \text{rank}([\mathbf{A} | \mathbf{b}])$，则该系统是一致的（即，它至少有一个解）。 2. **Inconsistent System**: If $\text{rank}(\mathbf{A}) \neq \text{rank}([\mathbf{A} | \mathbf{b}])$, the system is inconsistent (i.e., it has no solutions). - **不一致性系统**：如果 $\text{rank}(\mathbf{A}) \neq \text{rank}([\mathbf{A} | \mathbf{b}])$，则该系统是不一致的（即，它没有解）。 3. **Unique Solution**: If $\text{rank}(\mathbf{A}) = \text{rank}([\mathbf{A} | \mathbf{b}]) = n$ (where $n$ is the number of unknowns), the system has a unique solution. - **唯一解**：如果 $\text{rank}(\mathbf{A}) = \text{rank}([\mathbf{A} | \mathbf{b}]) = n$（其中 $n$ 是未知数的数量），则该系统有唯一解。 4. **Infinite Solutions**: If $\text{rank}(\mathbf{A}) = \text{rank}([\mathbf{A} | \mathbf{b}]) < n$, the system has infinitely many solutions. - **无限多解**：如果 $\text{rank}(\mathbf{A}) = \text{rank}([\mathbf{A} | \mathbf{b}]) < n$，则该系统有无限多个解。