*Linear Algebra, What you need to know* is an introductory book on Linear Algebra covering the standard topics for a one-semester course along with many applications. The highlights of the book include many solved examples, a varied problem set at the end of each chapter and theorems with proofs. The author provides a list of key sections to be covered in a one-semester course which is helpful to the instructor. The solved examples in each chapter illustrate definitions and theorems and explain how to solve problems of a computational nature. After explaining one or two examples in detail, the routine verifications in other examples are left to the reader. For the interested reader, proofs of theorems are provided as well. The exposition is concise in some technical proofs and in applications.

The organization of the course material differs a little from some standard textbooks. The notions of subspaces in \(\mathbf{R}^{n} \) and bases are covered earlier in Chapter 2 before the notion of a general vector space is covered and before the chapter on matrix algebra. Thus the concepts of row space, null space and column space of a matrix, the rank theorem, and coordinate systems are introduced much earlier in the course.

In the first chapter, the author explains an interesting way to check whether the solution to a system is correct by examining relationships between columns of the reduced row echelon form and that of the coefficient matrix. This can help to deepen understanding of the process of row reduction. The chapter on determinants has a section devoted to an alternate way of defining determinants using the sign function. The author uses this alternate definition to prove some of the properties of determinants. This section is identified as optional.

The concept of vector spaces is covered in Chapter 5. However, references to vector spaces are made very early in the book. The first reference to \(\mathbf{R}^{n}\) as a vector space is made when properties of vectors are studied in Chapter 1. Similarly, it is observed that \(\mathbf{R}^{m \times n}\) is a vector space over \(\mathbf{R}\) when studying matrix algebra in Chapter 3. These observations help students connect the concept between chapters.

The proofs in the last chapter on orthogonality are technical and concise but the process of orthogonal projection is introduced in an understandable way by first explaining the notion of distance between a vector and a subspace. Examples of finding an orthonormal basis using the Gram Schmidt process and finding the QR factorization of a matrix and its application to least-squares solutions are explained well. The last topic on singular value decomposition is also explained with examples. One other difference between this book and some standard textbooks on linear algebra is that the topic of complex eigenvalues is covered in this book. Theorems in the last chapter on orthogonality are proven for vectors over the field of real or complex numbers. The basic properties of complex numbers are covered in detail in the appendix.

The problem set at the end of each chapter includes a wide range of exercises ranging from routine calculations to problems that promote deeper understanding. For instance, in Chapter 1, finding constants so that \(ax + by = c\) is either parallel or identical to a given line or intersects the given line at a single point enhances understanding of the nature of solutions of linear systems and gives a geometrical interpretation for the solutions. This type of problem where one is expected to find constants so that a system or a matrix satisfies certain conditions appears throughout the text in various contexts such as linear independence, invertibility, determinants etc. An interesting exercise in the section on determinants involves finding determinants of other matrices by identifying the elementary row operations used to get from the matrix whose determinant is given to the matrix in the exercise. Some new ideas such as the Vandermonde matrix and its properties are given in the exercises as well. In Chapter 2, there are many exercises on computing bases for the row space, column space and null space of a matrix which provide good practice. All chapters include true or false exercises and in some chapters there are a few proof writing exercises or problems where students are expected to construct matrices satisfying certain properties. The proof of uniqueness of the reduced row echelon form is outlined in the exercises in Chapter 2. These different types of problems help to enhance understanding and clear misconceptions. The chapters on vector spaces and linear transformations have a variety of examples and exercises that show how these concepts generalize the notions of \(\mathbf{R}^n\) and linear maps between \(\mathbf{R}^n\) and \(\mathbf{R}^m\).

This book encourages familiarity with technology that can be used to check solutions to problems as well as for applications. There are footnotes on several pages in the book with syntax for using Python, Sage or Wolfram Alpha for a particular calculation. In addition, the author provides pseudocodes for the Gaussian Elimination Algorithm, and the LU and QR factorizations of a matrix.

Concise descriptions of many applications are provided throughout the book. These applications portray a vivid picture of the applicability of the subject of Linear Algebra. Perhaps a list of references could be provided for some applications for the interested reader. There are a few typos in the book in statements of theorems, or definitions which instructors can identify. A very limited set of answers are provided for exercise sets. For a quick check, it would be helpful if final answers provided to more problems in the book. An instructor solutions manual would be very helpful.

In summary, this book is an excellent reference for teaching a one-semester course in undergraduate Linear Algebra. There are many proofs that would be accessible to undergraduates. The author has explained various proof techniques in an interesting way, first with non-mathematical examples and then with mathematical examples at the end of the book. This book can therefore serve as a textbook for undergraduates who have the maturity and penchant for proof writing.

Hema Gopalakrishnan is associate professor of mathematics at Sacred Heart University and leads the Fairfield County Math Teachersâ€™ Circle in Connecticut.