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.

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

Rooted in a pedagogically successful problem-solving approach to linear algebra, the present work fills a gap in the literature that is sharply divided between elementary texts and books that are too advanced to appeal to a wide audience. It 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.

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 with Applications" 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.

Preface.- Linear Phenomena and Euclidean Spaces of Small Dimension.- Concrete Vector Spaces.- Vector Spaces and Subspaces.- Linear Transformations.- More Matrix Algebra and Determinants.- General Theory of Linear Equations.- Eigenvectors.- Orthogonality.- Forms.- Vector Spaces over Finite Fields.- Appendix A: Complex Numbers.- Appendix B: Polynomials over Complex Numbers.- References.- Index.