Essential Linear Algebra with Applications

A Problem-Solving Approach


Rooted in a pedagogically successful problem-solving approach to linear algebra, this work fills a gap in the literature that is sharply divided between, on the one end, elementary texts with only limited exercises and examples, and, at the other extreme, books too advanced in prerequisites and too specialized in focus to appeal to a wide audience. Instead, "Essential Linear Algebra" clearly develops the theoretical foundations of vector spaces, linear equations, matrix algebra, eigenvectors, and orthogonality, while simultaneously emphasizing applications and connections to fields such as biology, economics, computer graphics, electrical engineering, cryptography, and political science.

Key features:

* Intertwined discussion of linear algebra and geometry, giving readers a solid understanding of both topics and the relationship between them

* Each section starts with a concise overview of important concepts in results, followed by a selection of fully-solved problems

* Example-driven exposition

* Over 500 problems (roughly half include complete solutions) that are carefully selected for instructive appeal, elegance, and theoretical importance

* Two or more solutions provided to many of the problems; paired solutions range from step-by-step, elementary methods whose purpose is to strengthen basic comprehension to more sophisticated, powerful proofs to challenge advanced readers

* Appendices with review material on complex variables

Ideal as an introduction to linear algebra, the extensive exercises and well-chosen applications also make this text suitable for advanced courses at the junior or senior undergraduate level. Furthermore, it can serve as a colorful supplementary problem book, reference, or self-study manual for professional scientists and mathematicians. Complete with bibliography and index, "Essential Linear Algebra" is a natural bridge between pure and applied mathematics and the natural and social sciences, appropriate for any student or researcher who needs a strong footing in the theory, problem-solving, and model-building that are the subject’s hallmark.


  • Springer New York; October 2014
  • ISBN 9780817646363
  • Read online, or download in secure PDF format
  • Title: Essential Linear Algebra with Applications
  • Author: Titu Andreescu
  • Imprint: Birkhäuser

In The Press

“This book gives a rigorous introduction to linear algebra from a mathematical point of view. … The book presents a ‘problem-based’ approach, i.e., the author gives the precise theory to permit the reader to solve many different problems. … The book contains many well-chosen examples and exercises. Also, many problems are completely solved.” (Julio Benítez, Mathematical Reviews, June, 2015)

“The present book, as its title indicates, introduces the essential concepts of linear algebra and considers some of its applications. … the book is very suitable for students who are going to prepare for competitions and Olympiads. Moreover, the solved problems make it a very good problem book, with nice ideas, for anybody working with linear algebra. I recommend this book for students, and also for classroom use by teachers.” (Mehdi Hassani, MAA Reviews, April, 2015)

“The present book is a wellcomed addition at the present literature. The approach is elementray, however still rigorous and detailed. It covers the standard topics on linear algebra taught in a two-semester course, from elementray matrix theory to canonical forms and bilinear forms. … I would be happy to teach linear algebra from this book and ask students to work on problems from it.” (A. Arvanitoyeorgos, zbMATH 1309.15001, 2015)

About The Author

Titu Andreescu is an Associate Professor of Mathematics at the University of Texas at Dallas.