*Linear Algebra* by Nair and Singh introduces the student to the fundamental concepts and main results of linear algebra, which form the backbone of analysis, applied mathematics, and algebra. Designed to be an undergraduate text, this book is not for the faint of heart – it is a book for mathematics, science, and engineering majors providing students with a rigorous approach to the subject. The text is presented in seven chapters ranging from vector spaces, linear transformations, best approximations in inner product spaces, eigenvalues and eigenvectors, block diagonalisation, triangularisation, Jordan form, singular value decomposition, polar decomposition, and spectral representation. The problems at the end of each section motivate the student to think deeply about the concepts, theorems, and proofs presented.

Depending on the book you choose for your Linear Algebra course, some will start with systems of equations and make connections to the augmented matrix and others will start with matrix algebra. This text starts with Vector Spaces with the goals to help the student develop abstract thinking, carry out formal and rigorous arguments, and have students realize that matrices are not a static array of numbers but as a dynamic maps acting on vectors. The proofs presented in this text can be considered non-conventional. An example of this is that the proofs of linear transformations are presented first followed by the results on matrices.

The exercises presented in the text are theoretically based and motivate the student to recall concepts presented in the sections. Mathematics Educators would welcome this book as it prompts active learning and is a great resource for group projects or even competitions. For example, on page 75, section 2.3: Isomorphisms, exercise 7 states:

Let T: V ➔ W be a linear transformation, where V is a finite-dimensional vector space. Prove that the quotient space V/N(T) is isomorphic to R(T). Then deduce the rank-nullity theorem.

The very next problem, #8, asks students to look at surjective and injective linear transformations but also asks a follow-up question, “What could go wrong if the spaces are infinite dimensional?” It is these types of questions that help students not only to make connections between the concepts but also help the students gain a deeper thought and appreciation of Linear Algebra.

I mentioned at the start of this review that while the fundamental concepts of linear algebra are presented, this book is directed to junior or senior level undergraduate students. It would be to the student’s advantage to take Discrete Mathematics or some introduction to proof writing course before picking up this book, as it is heavily proofs-based. However, it is a very well written book. It is clear that Nair and Singh put a lot of work into the text to make the concepts elaborate and lively at the same time. This book can build the confidence of a student majoring in mathematics, science, or engineering by building their critical thinking skills and problem-solving skills – not to mention practice with writing proofs. I would highly consider reviewing this book for a linear algebra course you would be teaching in a future semester.

Peter Olszewski is a Mathematics Lecturer at The Pennsylvania State University, The Behrend College, an editor for Larson Texts, Inc. in Erie, PA, and is the 362nd Chapter Advisor of the Pennsylvania Alpha Beta Chapter of Pi Mu Epsilon. His Research fields are in mathematics education, Cayley Color Graphs, Markov Chains, and mathematical textbooks. He can be reached at pto2@psu.edu. Webpage: www.personal.psu.edu/pto2. Outside of teaching and textbook editing, he enjoys playing golf, playing guitar and bass, reading, gardening, traveling, and painting landscapes.