Linear Algebra for New Engineering

新工科线性代数是为大学人工智能等新工科专业开设的一门重要的数学基础必修课程。课程包含四个部分：第一部分介绍向量的基本运算以及最常见的线性空间——欧氏空间，并在欧氏空间中通过几何方法引入行列式的概念；第二部分介绍矩阵理论以及线性方程组的系统解法，通过贴近生活的实例具象化地解释线性代数的核心功能；第三部分介绍如何从线性映射的角度重新理解线性系统，讲解内积、正交性、特征值、奇异值、对角化等线性代数的重要概念以及图和网络、马尔科夫过程、三维点云融合、图像数据压缩等大量应用实例；第四部分介绍一般的线性空间特别是无限维线性空间，在无限维线性空间理论的基础上解释傅里叶分析这一工程学利器的原理和应用，并通过识别与分类问题展示线性代数在人工智能里的重要作用。通过学习本课程，学生将掌握人工智能等新工科所必需的线性代数理论，培养抽象思维、逻辑推理和解决实际问题的综合能力，为今后的学习和工作打下必要的数学基础。

Linear Algebra for New Engineering is an important compulsory foundation mathematics course designed for university majors such as Artificial Intelligence and other emerging engineering disciplines. The course consists of four parts. The first part introduces the basic operations of vectors and the most common linear space—the Euclidean space, and explains the concept of determinants in Euclidean space using geometric methods. The second part covers matrix theory and systematic methods for solving linear equations, illustrating the core functions of linear algebra through concrete examples drawn from real-life contexts. The third part demonstrates how to reinterpret linear systems from the perspective of linear mappings. It explains key concepts of linear algebra such as inner products, orthogonality, eigenvalues, singular values, and diagonalization, while also presenting numerous application cases including graphs and networks, Markov processes, 3D point cloud fusion, and image data compression. The fourth part introduces general linear spaces, especially infinite-dimensional linear spaces. Building on the theory of infinite-dimensional spaces, it explains the principles and applications of Fourier analysis—an essential tool in engineering—and showcases the significant role of linear algebra in Artificial Intelligence through tasks such as recognition and classification. By completing this course, students will master the linear algebraic theories essential for Artificial Intelligence and other New Engineering disciplines, develop abstract thinking, logical reasoning, and comprehensive problem-solving skills, and lay the necessary mathematical foundation for future study and work.