Advanced Linear Algebra and Artificial Intelligence: An Interactive Mathematics Textbook for Data Science and AI

Title: Advanced Linear Algebra and Artificial Intelligence: An Interactive Mathematics Textbook for Data Science and AI Original Author/Maintainer: wanghemath (Department of Mathematics, Northeastern University) Source Platform: GitHub Project Link: https://github.com/wanghemath/Book-AdvancedLinearAlgebraAI Online Reading: https://wanghemath.github.io/Book-AdvancedLinearAlgebraAI/ Release Time: June 2026

Core Introduction: This open-source interactive textbook launched by the Department of Mathematics at Northeastern University aims to bridge the gap between linear algebra theory and AI/data science applications. It extends abstract linear algebra theory to modern data science and AI fields, covering a complete knowledge system from basic matrix operations to SVD, PCA, neural networks, and matrix calculus. The textbook adopts an interactive design combined with Python practice to help learners connect theory and application.

In today's era of rapid AI development, many practitioners focus on using tools like PyTorch and TensorFlow but ignore the mathematical foundation behind them. Linear algebra is precisely the underlying language supporting modern machine learning and data science.

The core philosophy of this textbook is: Linear algebra is the language that describes structure, computation, geometry, data, and modern intelligent systems. It is not a traditional textbook but a journey from concrete computation to abstract theory, ultimately focusing on cutting-edge applications in AI and data science, aiming to help learners understand the mathematical essence behind the tools.

The uniqueness of the textbook lies in its design philosophy:

1. Integration of Four Perspectives:
   • Algebraic perspective: Focuses on basic concepts such as equations, matrices, linear transformations, and eigenvalues;
   • Geometric perspective: Introduces geometric meanings like dimension, orthogonality, and projection;
   • Computational perspective: Covers efficient algorithm implementations like LU decomposition, SVD, and FFT;
   • Application perspective: Demonstrates practical applications such as least squares method, PCA, Markov chains, and neural networks.
2. Story-Driven Learning: Each chapter starts with a practical problem (e.g., "When does a system of linear equations have no solution, a unique solution, or infinitely many solutions?"), guiding learners to build their knowledge system by solving problems instead of passively accepting definitions.

The textbook consists of 27 chapters, forming a complete learning path:

• Basics Section (Chapters 1-6): Basic content such as linear systems, matrix algebra, vector spaces, subspaces, linear mappings, and determinants;
• Core Section (Chapters 7-13): Core theories like eigenvalues/eigenvectors, diagonalization, inner product spaces, QR decomposition, and least squares method;
• Application Section (Chapters 14-27): Cutting-edge applications including SVD, PCA, wavelet transform, tensor product, spectral graph theory, and matrix calculus, covering fields like AI, signal processing, and optimization.

The interactive design of the textbook makes learning more engaging:

• Each chapter includes a story-driven opening, rigorous definitions and theorems, and runnable Python code examples;
• Provides Quarto-format experiment pages, Jupyter Notebook practice environments, and AI-assisted activities;
• Supports expandable/collapsible proofs and solutions to meet learning needs of different depths;
• Helps learners transform theory into practical skills through numerical experiments and challenging problems.

In the era of deep learning, many people fall into the trap of being "parameter-tuning engineers"—only knowing how to call APIs but not understanding the underlying principles. This textbook emphasizes:

• Matrices are not just arrays; they are transformations, models, projections, or neural network layers;
• Decompositions are not just formulas; they are ways to reveal the hidden structure of data.

Understanding these deep connections allows one to draw inferences from one case to another when facing new problems and design more elegant solutions (e.g., optimizing CV models, improving recommendation systems). A solid foundation in linear algebra is the core competitiveness of AI/ML practitioners.

Target Audience:

• Graduate students in applied mathematics/statistics;
• Data science practitioners who want to solidify their mathematical foundation;
• Machine learning engineers who want to understand algorithm principles;
• Researchers interested in the intersection of mathematics and AI.

Learning Suggestions:

1. Read the first 13 chapters in order to build a solid theoretical foundation;
2. Choose the last 14 chapters based on interests to delve into specific application areas;
3. Complete the Python experiments in each chapter to deepen understanding through hands-on practice;
4. Reinterpret familiar ML algorithms (e.g., PCA, neural network backpropagation) using the knowledge learned.