参考：Introduction to Applied Linear Algebra -- Vectors, Matrices, and Least Squares
Stephen Boyd and Lieven Vandenberghe
Website: https://web.stanford.edu/~boyd/vmls/
基底展开（Expansion in a Basis）：概念、推导与应用
1. 基本概念
在线性代数中，"基底展开"指的是用一组线性无关的向量基底来表示一个向量。在一个线性空间中，基底是描述空间中任意向量的基本单位。给定一组基底，空间中的任何向量都可以唯一地表示为这些基底的线性组合。
设 ( A = [ a 1 , a 2 , ... , a n ] A = [\mathbf{a}_1, \mathbf{a}_2, \dots, \mathbf{a}_n] A=[a1,a2,...,an] ) 是一个 ( n × n n \times n n×n ) 的方阵，其中每一列 ( a i \mathbf{a}_i ai ) 是 ( R n \mathbb{R}^n Rn ) 中的一个向量，并且它们线性无关。这意味着 ( A A A ) 的列向量构成了 ( R n \mathbb{R}^n Rn ) 的一个基底。对于任意一个向量 ( b ∈ R n \mathbf{b} \in \mathbb{R}^n b∈Rn )，存在唯一的向量 ( x ∈ R n \mathbf{x} \in \mathbb{R}^n x∈Rn )，使得：
A x = b . A \mathbf{x} = \mathbf{b}. Ax=b.
在这个公式中，向量 ( x = [ x 1 , x 2 , ... , x n ] T \mathbf{x} = [x_1, x_2, \dots, x_n]^T x=[x1,x2,...,xn]T ) 表示 ( b \mathbf{b} b ) 在基底 ( { a 1 , a 2 , ... , a n } \{\mathbf{a}_1, \mathbf{a}_2, \dots, \mathbf{a}_n\} {a1,a2,...,an} ) 下的系数（或坐标）。
2. 推导过程
（1）线性组合的定义
假设向量 ( b \mathbf{b} b ) 可以由基底 ( { a 1 , a 2 , ... , a n } \{\mathbf{a}_1, \mathbf{a}_2, \dots, \mathbf{a}_n\} {a1,a2,...,an} ) 表示为线性组合：
b = x 1 a 1 + x 2 a 2 + ⋯ + x n a n , \mathbf{b} = x_1 \mathbf{a}_1 + x_2 \mathbf{a}_2 + \cdots + x_n \mathbf{a}_n, b=x1a1+x2a2+⋯+xnan,
其中 ( x 1 , x 2 , ... , x n x_1, x_2, \dots, x_n x1,x2,...,xn ) 是待求的系数。
可以用矩阵表示这个关系：
A x = b , A \mathbf{x} = \mathbf{b}, Ax=b,
其中：
- ( A = [ a 1 , a 2 , ... , a n ] A = [\mathbf{a}_1, \mathbf{a}_2, \dots, \mathbf{a}_n] A=[a1,a2,...,an] ) 是一个 ( n × n n \times n n×n ) 的基底矩阵；
- ( x = [ x 1 , x 2 , ... , x n ] T \mathbf{x} = [x_1, x_2, \dots, x_n]^T x=[x1,x2,...,xn]T ) 是系数向量；
- ( b \mathbf{b} b ) 是目标向量。
（2）唯一解的存在性
由于 ( a 1 , a 2 , ... , a n \mathbf{a}_1, \mathbf{a}_2, \dots, \mathbf{a}_n a1,a2,...,an ) 是线性无关的，矩阵 ( A A A ) 的列向量构成一个基底，因此 ( A A A ) 是可逆矩阵。对方程 ( A x = b A \mathbf{x} = \mathbf{b} Ax=b )：
x = A − 1 b . \mathbf{x} = A^{-1} \mathbf{b}. x=A−1b.
这说明对于任意给定的 ( b \mathbf{b} b )，都存在唯一的系数向量 ( x \mathbf{x} x ) 使得 ( A x = b A \mathbf{x} = \mathbf{b} Ax=b )。
（3）几何意义
系数向量 ( x \mathbf{x} x ) 给出了 ( b \mathbf{b} b ) 在基底 ( { a 1 , a 2 , ... , a n } \{\mathbf{a}_1, \mathbf{a}_2, \dots, \mathbf{a}_n\} {a1,a2,...,an} ) 下的坐标，即 ( b \mathbf{b} b) 在该基底中的"展开"。
3. 示例
示例 1：二维空间的基底展开
假设二维空间中的基底为：
a 1 = [ 1 0 ] , a 2 = [ 1 1 ] . \mathbf{a}_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \quad \mathbf{a}_2 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}. a1=[10],a2=[11].
基底矩阵为：
A = [ 1 1 0 1 ] . A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}. A=[1011].
目标向量为：
b = [ 3 2 ] . \mathbf{b} = \begin{bmatrix} 3 \\ 2 \end{bmatrix}. b=[32].
我们希望找到 ( x = [ x 1 x 2 ] \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} x=[x1x2] )，使得：
A x = b . A \mathbf{x} = \mathbf{b}. Ax=b.
即：
1 1 0 1 \] \[ x 1 x 2 \] = \[ 3 2 \] . \\begin{bmatrix} 1 \& 1 \\\\ 0 \& 1 \\end{bmatrix} \\begin{bmatrix} x_1 \\\\ x_2 \\end{bmatrix} = \\begin{bmatrix} 3 \\\\ 2 \\end{bmatrix}. \[1011\]\[x1x2\]=\[32\]. 解方程： 1. 展开为： { x 1 + x 2 = 3 , x 2 = 2. \\begin{cases} x_1 + x_2 = 3, \\\\ x_2 = 2. \\end{cases} {x1+x2=3,x2=2. 2. 解得： x 1 = 1 , x 2 = 2. x_1 = 1, \\quad x_2 = 2. x1=1,x2=2. 因此，向量 ( b \\mathbf{b} b ) 的基底展开为： b = 1 ⋅ a 1 + 2 ⋅ a 2 . \\mathbf{b} = 1 \\cdot \\mathbf{a}_1 + 2 \\cdot \\mathbf{a}_2. b=1⋅a1+2⋅a2. ###### 示例 2：编程实现 用 Python 实现上述计算： ```python import numpy as np # 定义基底矩阵 A 和目标向量 b A = np.array([[1, 1], [0, 1]]) b = np.array([3, 2]) # 求解 x x = np.linalg.solve(A, b) print("The coefficients vector x is:", x) ``` 输出： ```c The coefficients vector x is: [1. 2.] ``` *** ** * ** *** ##### 4. 应用场景 基底展开在数学、物理、工程和计算机科学中都有广泛的应用。以下是一些常见的场景： ###### （1）计算机图形学 在图形变换中，通常需要将点的坐标从一个基底转换到另一个基底。例如，将世界坐标系中的点转换到相机坐标系。 ###### （2）数据科学和机器学习 在主成分分析（PCA）中，数据向量被投影到一组正交基上，这些基是数据协方差矩阵的特征向量。 ###### （3）量子力学 量子态通常表示为某个基底下的展开。例如，一个量子态可以用希尔伯特空间中的基向量展开。 ###### （4）信号处理 在傅里叶分析中，信号被展开为一组正交基（傅里叶基）的线性组合，用于频率分析。 *** ** * ** *** ##### 5. 总结 基底展开是线性代数中的一个重要概念，它提供了一种用基底来唯一表示向量的方法。通过基底展开，可以将复杂的向量操作简化为基底下的系数运算。在实际应用中，基底展开不仅具有深刻的理论意义，还在科学计算、机器学习、物理建模等领域发挥了重要作用。 在编程中，可以利用数值计算工具（如 Python 的 NumPy 或 MATLAB）快速实现基底展开的相关计算，方便解决实际问题。 ### 英文版 #### Expansion in a Basis: Concept, Derivation, and Applications ##### 1. Concept In linear algebra, **expansion in a basis** refers to expressing a vector in a vector space as a linear combination of a set of basis vectors. A basis is a set of linearly independent vectors that span the entire vector space. With respect to a given basis, any vector in the space can be uniquely represented as a weighted sum of the basis vectors. Let ( A = \[ a 1 , a 2 , ... , a n \] A = \[\\mathbf{a}_1, \\mathbf{a}_2, \\dots, \\mathbf{a}_n\] A=\[a1,a2,...,an\] ) be an ( n × n n \\times n n×n ) square matrix whose columns ( a 1 , a 2 , ... , a n \\mathbf{a}_1, \\mathbf{a}_2, \\dots, \\mathbf{a}_n a1,a2,...,an ) are linearly independent and form a basis of ( R n \\mathbb{R}\^n Rn ). For any vector ( b ∈ R n \\mathbf{b} \\in \\mathbb{R}\^n b∈Rn ), there exists a unique vector ( x ∈ R n \\mathbf{x} \\in \\mathbb{R}\^n x∈Rn ) such that: A x = b . A \\mathbf{x} = \\mathbf{b}. Ax=b. Here, the vector ( x \\mathbf{x} x ) contains the coefficients of ( b \\mathbf{b} b ) when expanded in the basis ( { a 1 , a 2 , ... , a n } \\{\\mathbf{a}_1, \\mathbf{a}_2, \\dots, \\mathbf{a}_n\\} {a1,a2,...,an} ). *** ** * ** *** ##### 2. Derivation ###### (1) Linear Combination If a vector ( b \\mathbf{b} b ) can be expressed as a linear combination of the basis vectors ( { a 1 , a 2 , ... , a n } \\{\\mathbf{a}_1, \\mathbf{a}_2, \\dots, \\mathbf{a}_n\\} {a1,a2,...,an} ), then: b = x 1 a 1 + x 2 a 2 + ⋯ + x n a n , \\mathbf{b} = x_1 \\mathbf{a}_1 + x_2 \\mathbf{a}_2 + \\cdots + x_n \\mathbf{a}_n, b=x1a1+x2a2+⋯+xnan, where ( x 1 , x 2 , ... , x n x_1, x_2, \\dots, x_n x1,x2,...,xn ) are the coefficients to be determined. This can be rewritten in matrix form as: A x = b , A \\mathbf{x} = \\mathbf{b}, Ax=b, where: * ( A = \[ a 1 , a 2 , ... , a n \] A = \[\\mathbf{a}_1, \\mathbf{a}_2, \\dots, \\mathbf{a}_n\] A=\[a1,a2,...,an\] ) is the basis matrix, * ( x = \[ x 1 , x 2 , ... , x n \] T \\mathbf{x} = \[x_1, x_2, \\dots, x_n\]\^T x=\[x1,x2,...,xn\]T ) is the coefficient vector, * ( b \\mathbf{b} b ) is the target vector. ###### (2) Existence and Uniqueness Since the columns of ( A A A ) are linearly independent, the matrix ( A A A ) is invertible. Thus, for any ( b \\mathbf{b} b ), the equation ( A x = b A \\mathbf{x} = \\mathbf{b} Ax=b ) has a unique solution: x = A − 1 b . \\mathbf{x} = A\^{-1} \\mathbf{b}. x=A−1b. ###### (3) Geometric Interpretation The vector ( x \\mathbf{x} x ) represents the coordinates of ( b \\mathbf{b} b ) in the basis ( { a 1 , a 2 , ... , a n } \\{\\mathbf{a}_1, \\mathbf{a}_2, \\dots, \\mathbf{a}_n\\} {a1,a2,...,an} ). In other words, ( x \\mathbf{x} x ) gives the unique decomposition of ( b \\mathbf{b} b ) in terms of the basis vectors. *** ** * ** *** ##### 3. Examples ###### Example 1: Basis Expansion in 2D Let's consider a 2D space with the basis: a 1 = \[ 1 0 \] , a 2 = \[ 1 1 \] . \\mathbf{a}_1 = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}, \\quad \\mathbf{a}_2 = \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix}. a1=\[10\],a2=\[11\]. The basis matrix is: A = \[ 1 1 0 1 \] . A = \\begin{bmatrix} 1 \& 1 \\\\ 0 \& 1 \\end{bmatrix}. A=\[1011\]. Suppose the target vector is: b = \[ 3 2 \] . \\mathbf{b} = \\begin{bmatrix} 3 \\\\ 2 \\end{bmatrix}. b=\[32\]. We want to find ( x = \[ x 1 x 2 \] \\mathbf{x} = \\begin{bmatrix} x_1 \\\\ x_2 \\end{bmatrix} x=\[x1x2\] ) such that: A x = b . A \\mathbf{x} = \\mathbf{b}. Ax=b. Solving: 1. Expand the equation: \[ 1 1 0 1 \] \[ x 1 x 2 \] = \[ 3 2 \] . \\begin{bmatrix} 1 \& 1 \\\\ 0 \& 1 \\end{bmatrix} \\begin{bmatrix} x_1 \\\\ x_2 \\end{bmatrix} = \\begin{bmatrix} 3 \\\\ 2 \\end{bmatrix}. \[1011\]\[x1x2\]=\[32\]. This gives: { x 1 + x 2 = 3 , x 2 = 2. \\begin{cases} x_1 + x_2 = 3, \\\\ x_2 = 2. \\end{cases} {x1+x2=3,x2=2. 2. Solve for ( x 1 x_1 x1 ) and ( x 2 x_2 x2 ): x 2 = 2 , x 1 = 1. x_2 = 2, \\quad x_1 = 1. x2=2,x1=1. Thus, the vector ( b \\mathbf{b} b ) can be written as: b = 1 ⋅ a 1 + 2 ⋅ a 2 . \\mathbf{b} = 1 \\cdot \\mathbf{a}_1 + 2 \\cdot \\mathbf{a}_2. b=1⋅a1+2⋅a2. ###### Example 2: Python Implementation Here's how to compute the basis expansion in Python: ```python import numpy as np # Define the basis matrix A and the target vector b A = np.array([[1, 1], [0, 1]]) b = np.array([3, 2]) # Solve for x x = np.linalg.solve(A, b) print("The coefficients vector x is:", x) ``` Output: ```c The coefficients vector x is: [1. 2.] ``` *** ** * ** *** ##### 4. Applications ###### (1) Computer Graphics In computer graphics, basis expansion is used to transform points between coordinate systems, such as converting world coordinates to camera coordinates. ###### (2) Data Science and Machine Learning In Principal Component Analysis (PCA), data vectors are projected onto a set of orthogonal basis vectors (principal components) to reduce dimensionality while preserving variance. ###### (3) Quantum Mechanics In quantum mechanics, quantum states are represented as linear combinations of basis vectors in a Hilbert space. ###### (4) Signal Processing In Fourier analysis, signals are expressed as sums of sinusoidal basis functions, allowing analysis in the frequency domain. *** ** * ** *** ##### 5. Summary Expansion in a basis is a fundamental concept in linear algebra that allows any vector to be expressed uniquely in terms of a given set of basis vectors. This concept has profound implications in mathematics, physics, engineering, and computer science. The process of expanding a vector in a basis involves solving a system of linear equations, which can be efficiently computed using numerical tools like Python's `numpy` or MATLAB. Understanding this concept enables us to work more effectively with vector spaces and their applications in a wide range of fields. ### 后记 2024年12月20日14点34分于上海， 在GPT4o大模型辅助下完成。