线性映射（Linear Mapping）原理详解：机器学习中的数学基石

钟智强2025-11-23 14:42

摘要

线性映射是机器学习领域中最基础且至关重要的数学概念之一。作为向量空间之间的结构保持变换，线性映射为理解机器学习算法提供了坚实的数学基础。本文将深入探讨线性映射的核心原理、数学性质及其在实际应用中的实现。

1. 线性映射的数学定义与核心特性

1.1 形式化定义

设 VVV 和 WWW 是域 FFF 上的两个向量空间，映射 T:V→WT: V \rightarrow WT:V→W 被称为线性映射，当且仅当满足以下两个条件：

可加性 ：
T(u+v)=T(u)+T(v)∀u,v∈VT(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \quad \forall \mathbf{u}, \mathbf{v} \in VT(u+v)=T(u)+T(v)∀u,v∈V

齐次性 ：
T(cv)=cT(v)∀v∈V,∀c∈FT(c\mathbf{v}) = cT(\mathbf{v}) \quad \forall \mathbf{v} \in V, \forall c \in FT(cv)=cT(v)∀v∈V,∀c∈F

1.2 基本性质推导

  • 零向量保持 ：T(0V)=0WT(\mathbf{0}_V) = \mathbf{0}_WT(0V)=0W
  • 负向量保持 ：T(−v)=−T(v)T(-\mathbf{v}) = -T(\mathbf{v})T(−v)=−T(v)
  • 线性组合保持 ：T(c1v1+c2v2+⋯+ckvk)=c1T(v1)+c2T(v2)+⋯+ckT(vk)T(c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k) = c_1T(\mathbf{v}_1) + c_2T(\mathbf{v}_2) + \cdots + c_kT(\mathbf{v}_k)T(c1v1+c2v2+⋯+ckvk)=c1T(v1)+c2T(v2)+⋯+ckT(vk)
c 复制代码 
#include <stdio.h>
#include <stdlib.h>
#include <math.h>

typedef struct {
    double* data;
    int rows;
    int cols;
} Matrix;

Matrix create_matrix(int rows, int cols) {
    Matrix mat;
    mat.rows = rows;
    mat.cols = cols;
    mat.data = (double*)malloc(rows * cols * sizeof(double));
    return mat;
}

void free_matrix(Matrix mat) {
    free(mat.data);
}

void matrix_multiply(Matrix A, Matrix B, Matrix result) {
    for(int i = 0; i < A.rows; i++) {
        for(int j = 0; j < B.cols; j++) {
            double sum = 0.0;
            for(int k = 0; k < A.cols; k++) {
                sum += A.data[i * A.cols + k] * B.data[k * B.cols + j];
            }
            result.data[i * result.cols + j] = sum;
        }
    }
}

int main() {
    Matrix A = create_matrix(2, 2);
    A.data[0] = 2.0; A.data[1] = 1.0;
    A.data[2] = 1.0; A.data[3] = 3.0;
    
    Matrix v1 = create_matrix(2, 1);
    Matrix v2 = create_matrix(2, 1);
    v1.data[0] = 1.0; v1.data[1] = 2.0;
    v2.data[0] = 3.0; v2.data[1] = 1.0;
    
    Matrix v_sum = create_matrix(2, 1);
    v_sum.data[0] = v1.data[0] + v2.data[0];
    v_sum.data[1] = v1.data[1] + v2.data[1];
    
    Matrix result1 = create_matrix(2, 1);
    Matrix result2 = create_matrix(2, 1);
    Matrix temp1 = create_matrix(2, 1);
    Matrix temp2 = create_matrix(2, 1);
    
    matrix_multiply(A, v_sum, result1);
    matrix_multiply(A, v1, temp1);
    matrix_multiply(A, v2, temp2);
    
    result2.data[0] = temp1.data[0] + temp2.data[0];
    result2.data[1] = temp1.data[1] + temp2.data[1];
    
    printf("T(v1 + v2) = [%.2f, %.2f]\n", result1.data[0], result1.data[1]);
    printf("T(v1) + T(v2) = [%.2f, %.2f]\n", result2.data[0], result2.data[1]);
    
    free_matrix(A); free_matrix(v1); free_matrix(v2);
    free_matrix(v_sum); free_matrix(result1); free_matrix(result2);
    free_matrix(temp1); free_matrix(temp2);
    
    return 0;
}

2. 线性映射的矩阵表示与几何解释

2.1 矩阵表示理论

对于有限维向量空间，任何线性映射都可以用矩阵来表示。设 T:Rn→RmT: \mathbb{R}^n \rightarrow \mathbb{R}^mT:Rn→Rm 是线性映射，则存在矩阵 AAA 使得：
T(x)=AxT(\mathbf{x}) = A\mathbf{x}T(x)=Ax

c 复制代码 
#include <stdio.h>
#include <math.h>

#define PI 3.14159265358979323846

typedef struct {
    double x;
    double y;
} Point2D;

void apply_linear_transform(Point2D* points, int count, double matrix[2][2]) {
    for(int i = 0; i < count; i++) {
        double x_old = points[i].x;
        double y_old = points[i].y;
        points[i].x = matrix[0][0] * x_old + matrix[0][1] * y_old;
        points[i].y = matrix[1][0] * x_old + matrix[1][1] * y_old;
    }
}

void print_points(Point2D* points, int count, const char* label) {
    printf("%s: ", label);
    for(int i = 0; i < count; i++) {
        printf("(%.2f,%.2f) ", points[i].x, points[i].y);
    }
    printf("\n");
}

int main() {
    Point2D triangle[] = {{0,0}, {1,0}, {0.5,1}, {0,0}};
    int count = 4;
    
    double scale_matrix[2][2] = {{2.0,0.0}, {0.0,1.5}};
    double angle = PI/4;
    double rotation_matrix[2][2] = {{cos(angle),-sin(angle)}, {sin(angle),cos(angle)}};
    double shear_matrix[2][2] = {{1.0,0.5}, {0.0,1.0}};
    
    Point2D scaled[4], rotated[4], sheared[4];
    
    for(int i=0; i<count; i++) scaled[i]=triangle[i];
    for(int i=0; i<count; i++) rotated[i]=triangle[i];
    for(int i=0; i<count; i++) sheared[i]=triangle[i];
    
    apply_linear_transform(scaled, count, scale_matrix);
    apply_linear_transform(rotated, count, rotation_matrix);
    apply_linear_transform(sheared, count, shear_matrix);
    
    print_points(triangle, count, "Original");
    print_points(scaled, count, "Scaled");
    print_points(rotated, count, "Rotated");
    print_points(sheared, count, "Sheared");
    
    return 0;
}

3. 核与像的空间理论

3.1 核心概念

核 ：
Ker(T)={v∈V∣T(v)=0}\text{Ker}(T) = \{\mathbf{v} \in V \mid T(\mathbf{v}) = \mathbf{0}\}Ker(T)={v∈V∣T(v)=0}

像 ：
Im(T)={T(v)∈W∣v∈V}\text{Im}(T) = \{T(\mathbf{v}) \in W \mid \mathbf{v} \in V\}Im(T)={T(v)∈W∣v∈V}

3.2 秩-零化度定理

dim(Ker(T))+dim(Im(T))=dim(V)\text{dim}(\text{Ker}(T)) + \text{dim}(\text{Im}(T)) = \text{dim}(V)dim(Ker(T))+dim(Im(T))=dim(V)

c 复制代码 
#include <stdio.h>
#include <stdlib.h>
#include <math.h>

#define EPSILON 1e-10

int matrix_rank(double** matrix, int rows, int cols) {
    int rank = 0;
    double** temp = (double**)malloc(rows * sizeof(double*));
    for(int i = 0; i < rows; i++) {
        temp[i] = (double*)malloc(cols * sizeof(double));
        for(int j = 0; j < cols; j++) {
            temp[i][j] = matrix[i][j];
        }
    }
    
    for(int col = 0; col < cols && rank < rows; col++) {
        int pivot_row = -1;
        for(int row = rank; row < rows; row++) {
            if(fabs(temp[row][col]) > EPSILON) {
                pivot_row = row;
                break;
            }
        }
        
        if(pivot_row == -1) continue;
        
        if(pivot_row != rank) {
            double* temp_ptr = temp[rank];
            temp[rank] = temp[pivot_row];
            temp[pivot_row] = temp_ptr;
        }
        
        double pivot = temp[rank][col];
        for(int j = col; j < cols; j++) {
            temp[rank][j] /= pivot;
        }
        
        for(int i = rank + 1; i < rows; i++) {
            double factor = temp[i][col];
            for(int j = col; j < cols; j++) {
                temp[i][j] -= factor * temp[rank][j];
            }
        }
        
        rank++;
    }
    
    for(int i = 0; i < rows; i++) {
        free(temp[i]);
    }
    free(temp);
    
    return rank;
}

int main() {
    int rows = 3, cols = 3;
    double** A = (double**)malloc(rows * sizeof(double*));
    for(int i = 0; i < rows; i++) {
        A[i] = (double*)malloc(cols * sizeof(double));
    }
    
    A[0][0] = 1.0; A[0][1] = 2.0; A[0][2] = 3.0;
    A[1][0] = 2.0; A[1][1] = 4.0; A[1][2] = 6.0;
    A[2][0] = 1.0; A[2][1] = 1.0; A[2][2] = 1.0;
    
    int rank = matrix_rank(A, rows, cols);
    int nullity = cols - rank;
    
    printf("Rank: %d\n", rank);
    printf("Nullity: %d\n", nullity);
    printf("Rank + Nullity: %d\n", rank + nullity);
    printf("Domain Dimension: %d\n", cols);
    
    for(int i = 0; i < rows; i++) {
        free(A[i]);
    }
    free(A);
    
    return 0;
}

4. 机器学习中的线性映射应用

4.1 基础线性模型

线性回归、逻辑回归等基础模型本质上是线性映射的应用。

c 复制代码 
#include <stdio.h>
#include <stdlib.h>

typedef struct {
    double** weights;
    double* bias;
    int input_dim;
    int output_dim;
} LinearModel;

LinearModel create_linear_model(int input_dim, int output_dim) {
    LinearModel model;
    model.input_dim = input_dim;
    model.output_dim = output_dim;
    
    model.weights = (double**)malloc(output_dim * sizeof(double*));
    for(int i = 0; i < output_dim; i++) {
        model.weights[i] = (double*)malloc(input_dim * sizeof(double));
        for(int j = 0; j < input_dim; j++) {
            model.weights[i][j] = 0.1;
        }
    }
    
    model.bias = (double*)malloc(output_dim * sizeof(double));
    for(int i = 0; i < output_dim; i++) {
        model.bias[i] = 0.0;
    }
    
    return model;
}

void free_linear_model(LinearModel model) {
    for(int i = 0; i < model.output_dim; i++) {
        free(model.weights[i]);
    }
    free(model.weights);
    free(model.bias);
}

void forward_pass(LinearModel model, double* input, double* output) {
    for(int i = 0; i < model.output_dim; i++) {
        output[i] = model.bias[i];
        for(int j = 0; j < model.input_dim; j++) {
            output[i] += model.weights[i][j] * input[j];
        }
    }
}

int main() {
    int input_size = 5;
    int output_size = 3;
    int batch_size = 2;
    
    LinearModel model = create_linear_model(input_size, output_size);
    
    double** inputs = (double**)malloc(batch_size * sizeof(double*));
    for(int i = 0; i < batch_size; i++) {
        inputs[i] = (double*)malloc(input_size * sizeof(double));
        for(int j = 0; j < input_size; j++) {
            inputs[i][j] = (i + j) * 0.1;
        }
    }
    
    double** outputs = (double**)malloc(batch_size * sizeof(double*));
    for(int i = 0; i < batch_size; i++) {
        outputs[i] = (double*)malloc(output_size * sizeof(double));
    }
    
    for(int i = 0; i < batch_size; i++) {
        forward_pass(model, inputs[i], outputs[i]);
        printf("Batch %d output: ", i);
        for(int j = 0; j < output_size; j++) {
            printf("%.3f ", outputs[i][j]);
        }
        printf("\n");
    }
    
    for(int i = 0; i < batch_size; i++) {
        free(inputs[i]);
        free(outputs[i]);
    }
    free(inputs);
    free(outputs);
    free_linear_model(model);
    
    return 0;
}

5. 奇异值分解与线性映射

c 复制代码 
#include <stdio.h>
#include <stdlib.h>
#include <math.h>

void matrix_print(double** A, int rows, int cols) {
    for(int i = 0; i < rows; i++) {
        for(int j = 0; j < cols; j++) {
            printf("%8.4f ", A[i][j]);
        }
        printf("\n");
    }
}

void svd_2x2(double A[2][2], double U[2][2], double S[2], double Vt[2][2]) {
    double a = A[0][0], b = A[0][1], c = A[1][0], d = A[1][1];
    
    double M = a*a + b*b + c*c + d*d;
    double sqrt_M = sqrt(M);
    double sqrt_M2_4ad_4bc = sqrt(M*M - 4*(a*d - b*c)*(a*d - b*c));
    
    S[0] = sqrt((M + sqrt_M2_4ad_4bc) / 2);
    S[1] = sqrt((M - sqrt_M2_4ad_4bc) / 2);
    
    double theta = 0.5 * atan2(2*(a*b + c*d), a*a + c*c - b*b - d*d);
    double phi = 0.5 * atan2(2*(a*c + b*d), a*a + b*b - c*c - d*d);
    
    U[0][0] = cos(phi); U[0][1] = -sin(phi);
    U[1][0] = sin(phi); U[1][1] = cos(phi);
    
    Vt[0][0] = cos(theta); Vt[0][1] = sin(theta);
    Vt[1][0] = -sin(theta); Vt[1][1] = cos(theta);
}

int main() {
    double A[2][2] = {{3,1}, {1,3}};
    double U[2][2], Vt[2][2], S[2];
    
    svd_2x2(A, U, S, Vt);
    
    printf("Original matrix A:\n");
    printf("%.4f %.4f\n", A[0][0], A[0][1]);
    printf("%.4f %.4f\n", A[1][0], A[1][1]);
    
    printf("\nSingular values: %.4f, %.4f\n", S[0], S[1]);
    
    printf("\nLeft singular vectors U:\n");
    printf("%.4f %.4f\n", U[0][0], U[0][1]);
    printf("%.4f %.4f\n", U[1][0], U[1][1]);
    
    printf("\nRight singular vectors Vt:\n");
    printf("%.4f %.4f\n", Vt[0][0], Vt[0][1]);
    printf("%.4f %.4f\n", Vt[1][0], Vt[1][1]);
    
    return 0;
}

6. 数值稳定性与性能优化

c 复制代码 
#include <stdio.h>
#include <stdlib.h>
#include <math.h>

typedef struct {
    double** data;
    int size;
} SquareMatrix;

SquareMatrix create_square_matrix(int size) {
    SquareMatrix mat;
    mat.size = size;
    mat.data = (double**)malloc(size * sizeof(double*));
    for(int i = 0; i < size; i++) {
        mat.data[i] = (double*)malloc(size * sizeof(double));
    }
    return mat;
}

void free_square_matrix(SquareMatrix mat) {
    for(int i = 0; i < mat.size; i++) {
        free(mat.data[i]);
    }
    free(mat.data);
}

double matrix_condition_number(SquareMatrix A) {
    double norm = 0.0;
    for(int i = 0; i < A.size; i++) {
        double row_sum = 0.0;
        for(int j = 0; j < A.size; j++) {
            row_sum += fabs(A.data[i][j]);
        }
        if(row_sum > norm) norm = row_sum;
    }
    return norm;
}

SquareMatrix ridge_regression(SquareMatrix A, double lambda) {
    SquareMatrix result = create_square_matrix(A.size);
    
    for(int i = 0; i < A.size; i++) {
        for(int j = 0; j < A.size; j++) {
            result.data[i][j] = A.data[i][j];
            if(i == j) {
                result.data[i][j] += lambda;
            }
        }
    }
    
    return result;
}

int main() {
    SquareMatrix ill_conditioned = create_square_matrix(2);
    ill_conditioned.data[0][0] = 1.0;
    ill_conditioned.data[0][1] = 1.0;
    ill_conditioned.data[1][0] = 1.0;
    ill_conditioned.data[1][1] = 1.0001;
    
    double cond_number = matrix_condition_number(ill_conditioned);
    printf("Condition number: %.2e\n", cond_number);
    
    SquareMatrix regularized = ridge_regression(ill_conditioned, 0.01);
    
    printf("Regularized matrix:\n");
    for(int i = 0; i < 2; i++) {
        for(int j = 0; j < 2; j++) {
            printf("%.6f ", regularized.data[i][j]);
        }
        printf("\n");
    }
    
    free_square_matrix(ill_conditioned);
    free_square_matrix(regularized);
    
    return 0;
}

结论

线性映射作为机器学习数学基础的核心构件，为理解复杂模型提供了统一的数学框架。通过掌握线性映射的理论性质和实际实现，开发者能够设计出更高效、稳定的机器学习系统。本文提供的C语言实现展示了线性映射在实践中的具体应用，为工程实践提供了可靠参考。

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