© 2014

Essential Linear Algebra with Applications

A Problem-Solving Approach


Table of contents

  1. Front Matter
    Pages i-x
  2. Titu Andreescu
    Pages 1-52
  3. Titu Andreescu
    Pages 53-83
  4. Titu Andreescu
    Pages 85-105
  5. Titu Andreescu
    Pages 107-148
  6. Titu Andreescu
    Pages 149-196
  7. Titu Andreescu
    Pages 197-235
  8. Titu Andreescu
    Pages 237-300
  9. Titu Andreescu
    Pages 339-375
  10. Titu Andreescu
    Pages 377-482
  11. Titu Andreescu
    Pages 483-490
  12. Back Matter
    Pages 491-491

About this book


This textbook provides a rigorous introduction to linear algebra in addition to material suitable for a more advanced course while emphasizing the subject’s interactions with other topics in mathematics such as calculus and geometry. A problem-based approach is used to develop the theoretical foundations of vector spaces, linear equations, matrix algebra, eigenvectors, and orthogonality.

Key features include:

• a thorough presentation of the main results in linear algebra along with numerous examples to illustrate the theory;

 • over 500 problems (half with complete solutions) carefully selected for their elegance and theoretical significance;

• an interleaved discussion of geometry and linear algebra, giving readers a solid understanding of both topics and the relationship between them.


Numerous exercises and well-chosen examples make this text suitable for advanced courses at the junior or senior levels. It can also serve as a source of supplementary problems for a sophomore-level course. 



Cayley-Hamilton theorem Determinant Diagonalizable Duality Linear maps Linear systems Matrices Minimal polynomial Quadratic form Vector space

Authors and affiliations

  1. 1.Natural Sciences and MathematicsUniversity of Texas at DallasRichardsonUSA

About the authors

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

Bibliographic information

Industry Sectors
Finance, Business & Banking


“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)