Linear algebra is one of the foundational pillars of modern mathematics and computing. Its concepts underpin a wide range of applied disciplines, including computer graphics, machine learning, data compression, network analysis, and engineering.
As data-driven methods continue to grow in prominence, a solid grounding in linear algebra has become increasingly important for students in mathematics, computer science, engineering, and the natural sciences.
The need for open, accessible, and pedagogically sound instructional materials has made openly licensed textbooks an increasingly valuable resource in higher education.
About the book
Understanding Linear Algebra by David Austin is a freely available, open-access textbook designed for first-year undergraduate students taking an introductory linear algebra course. The book does not assume prior knowledge of calculus and is structured to support both traditional lecture formats and active learning environments.
Each section begins with preview activities for students to complete before class and includes in-class activities designed for collaborative group work. The text integrates computational tools through embedded SageMath cells, enabling students to perform and verify linear algebraic operations throughout the reading.
The book has been endorsed by the Open Textbook Initiative at the American Institute of Mathematics and was recognized with the 2025 Daniel Solow Author’s Award by the Mathematical Association of America.
What you will learn
Readers will develop conceptual understanding of the core ideas in linear algebra, including how to set up and solve systems of linear equations, how to interpret geometric transformations using matrices, and how to analyze the structure of vector spaces.
The book builds toward an understanding of key decompositions such as eigenvalue decompositions and singular value decompositions, with attention to their real-world applications in areas such as image compression, Google PageRank, principal component analysis, and computer animation. Students will also develop computational fluency using SageMath, learning to apply software tools alongside theoretical reasoning.
Table of contents
- Chapter 1: Systems of equations
- Chapter 2: Vectors, matrices, and linear combinations
- Chapter 3: Invertibility, bases, and coordinate systems
- Chapter 4: Eigenvalues and eigenvectors
- Chapter 5: Linear algebra and computing
- Chapter 6: Orthogonality and Least Squares
- Chapter 7: Singular value decompositions
Book details
- Title: Understanding Linear Algebra
- Author(s): David Austin
- Main category: Mathematics
- Subcategory: Algebra
- Language: English
- License: Creative Commons Attribution 4.0 International (CC BY 4.0)
More books in: Algebra, Mathematics
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