MATH5110: A Graduate-level Applied Linear Algebra Textbook for Data Science and Artificial Intelligence

MATH5110Book is an open-source online textbook designed specifically for graduate students, focusing on applied linear algebra, matrix analysis, computation, and the mathematical foundations in data science and artificial intelligence. This textbook combines traditional mathematical theory with modern data science applications, providing a comprehensive knowledge system from theory to practice, and is a free, high-quality learning resource.

• Original Author/Maintainer: wanghemath
• Source Platform: GitHub
• Original Title: MATH5110Book
• Original Link: https://github.com/wanghemath/MATH5110Book
• Publication Date: 2026-05-31

Linear algebra is the mathematical cornerstone of artificial intelligence and data science. From weight matrix operations in neural networks to principal component analysis (PCA) in dimensionality reduction, from matrix factorization in recommendation systems to image transformations in computer vision, linear algebra is everywhere. For graduate students, understanding matrix analysis not only helps master algorithm principles but also provides theoretical support for model design and optimization.

The textbook covers core topics of graduate-level applied linear algebra: starting from basic matrix operations and vector space theory, it gradually deepens into matrix factorization, eigenvalue analysis, and numerical computation methods. It particularly emphasizes the combination of theory and application, with each concept accompanied by specific application scenarios in data science and AI.

Matrix analysis is key to understanding modern machine learning. In deep learning, neural network layers are essentially combinations of matrix transformations and nonlinear activations; the query-key-value computation in attention mechanisms is matrix multiplication, and understanding its mathematical essence helps improve the Transformer architecture. Mastering the properties of matrix operations can facilitate efficient architecture design and algorithm optimization.

The textbook covers core content of computational linear algebra: matrix condition numbers, numerical stability analysis, and iterative solution methods. For engineers dealing with large-scale data, these concepts help select appropriate algorithms and avoid numerical pitfalls. Especially in high-dimensional sparse data scenarios, numerical methods directly affect training efficiency and model performance.

As an open-source project, MATH5110Book provides free, high-quality resources, and its online format supports continuous updates to keep up with the development of the field. For researchers and engineers who want to deeply understand the mathematical principles of algorithms, this systematic textbook is an indispensable reference, and mastering these foundations will lay a solid groundwork for AI research and innovation.