Introduction to Linear and Matrix Algebra

1. Front Matter

   Pages i-xvi

   PDF

2. Vectors and Geometry

   • Nathaniel Johnston

   Pages 1-76

3. Linear Systems and Subspaces

   • Nathaniel Johnston

   Pages 77-233

4. Unraveling Matrices

   • Nathaniel Johnston

   Pages 235-401

5. Back Matter

   Pages 403-482

   PDF

This textbook emphasizes the interplay between algebra and geometry to motivate the study of linear algebra. Matrices and linear transformations are presented as two sides of the same coin, with their connection motivating inquiry throughout the book. By focusing on this interface, the author offers a conceptual appreciation of the mathematics that is at the heart of further theory and applications. Those continuing to a second course in linear algebra will appreciate the companion volume Advanced Linear and Matrix Algebra. 

Starting with an introduction to vectors, matrices, and linear transformations, the book focuses on building a geometric intuition of what these tools represent. Linear systems offer a powerful application of the ideas seen so far, and lead onto the introduction of subspaces, linear independence, bases, and rank. Investigation then focuses on the algebraic properties of matrices that illuminate the geometry of the linear transformations that they represent. Determinants, eigenvalues, and eigenvectors all benefit from this geometric viewpoint. Throughout, “Extra Topic” sections augment the core content with a wide range of ideas and applications, from linear programming, to power iteration and linear recurrence relations. Exercises of all levels accompany each section, including many designed to be tackled using computer software.

Introduction to Linear and Matrix Algebra is ideal for an introductory proof-based linear algebra course. The engaging color presentation and frequent marginal notes showcase the author’s visual approach. Students are assumed to have completed one or two university-level mathematics courses, though calculus is not an explicit requirement. Instructors will appreciate the ample opportunities to choose topics that align with the needs of each classroom, and the online homework sets that are available through WeBWorK.

• First course in linear algebra textbook
• Linear algebra textbook
• Matrix algebra textbook
• Linear algebra textbook WeBWorK
• Matrix algebra vs linear algebra
• Vector operations
• The dot product
• Matrix operations
• Linear transformation matrix
• Matrix inverses
• Subspace linear algebra
• Coordinate systems linear algebra
• Diagonalization linear algebra
• Graph theory introduction
• Linear systems
• Linear programming
• Power iteration PageRank
• Google PageRank algorithm
• Linear recurrence relations
• Geometric approach to linear algebra

“This book covers relatively limited and elementary ground in considerable detail. … It seems to me an excellent first course for undergraduates without a strong mathematical background … . International Centres such as those taking more ambitious CIE examinations would certainly find it useful. Above all, any schools where further mathematics is taught would be well advised to have a copy for departmental use; teachers who read it would feel more confident in their teaching, and I am sure they would enjoy the experience.” (Owen Toller, The Mathematical Gazette, Vol. 107 (568), March, 2023)

“The book is easy to read and requires almost no basic prior knowledge. … This is a good textbook on linear and matrix algebra. I would recommend this book to students who are learning linear algebra for the first time.” (Qing-Wen Wang, zbMATH 1481.15001, 2022)

• Mathematics and Computer Science, Mount Allison University, Sackville, Canada

  Nathaniel Johnston

Nathaniel Johnston is an Associate Professor of Mathematics at Mount Allison University in New Brunswick, Canada. His research makes use of linear algebra, matrix analysis, and convex optimization to tackle questions related to the theory of quantum entanglement. His companion volume, Advanced Linear and Matrix Algebra, is also published by Springer.

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