Matrix Theory and Linear Algebra

Matrix Theory and Linear Algebra is an introduction to linear algebra for students in the first or second year of university. The book contains enough material for a 2-semester course. Major topics of linear algebra are presented in detail, and many applications are given. Although it is not a proof-oriented book, proofs of most important theorems are provided.


1 Systems of linear equations
2 Vectors in ℝⁿ
3 Lines and planes in ℝⁿ
4 Matrices
5 Spans, linear independence, and bases in ℝⁿ
6 Linear transformations in ℝⁿ
7 Determinants
8 Eigenvalues, eigenvectors, and diagonalization
9 Vector spaces
10 Linear transformation of vector spaces
11 Inner product spaces
Appendix A Complex numbers

List of applications


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Matrix Theory and Linear Algebra is an open text, licensed under the Creative Commons CC BY 4.0 License. This means that it can be freely downloaded, printed, and shared (subject to some conditions that are spelled out in the license). The license also permits making changes. This is ideal for instructors who would like to add their own material, change notations, or add more examples or exercises. Please see the License for details. If you make revisions, please send them to me so that I can consider incorporating them in future versions of this book.

Reporting typos

Since this is an open text, typos can be easily fixed and an updated version posted online. It is my intention to fix all typos. If you find a typo (no matter how small), please report it to

Supplementary materials

Instructor resources

We have developed online homework assignments for this textbook using the MAA's WeBWorK system. We have also developed a set of tutorial worksheets with additional applications. To get these resources, instructors should contact

Source code

The LaTeX sources for this textbook are maintained as a GIT repository and can be found at


Matrix Theory and Linear Algebra in its current form was adapted, thoroughly revised, and extended by Peter Selinger for use at Dalhousie University. It is based on the earlier open text "A first course in linear algebra" by Lyryx Learning, which was in turns an adaptation of the open text "Elementary linear algebra" by Ken Kuttler. Please see inside the book for the full revision history.

Contributors include: Ken Kuttler, Ilijas Farah, Marie-Andrée B. Langlois, Peter Selinger, and the Lyryx Learning Team: Bruce Bauslaugh, Peter Chow, Nathan Friess, Stephanie Keyowski, Claude Laflamme, Martha Laflamme, Jennifer MacKenzie, Tamsyn Murnaghan, Bogdan Sava, Larissa Stone, Ryan Yee, and Ehsun Zahedi. Cover photography by Pok Rie.