1.1. Review by: A B Farnell.
The American Mathematical Monthly 76 (8) (1969), 958.

The text appears to be an excellent introduction to matrices and linear transformations. With aims of making the material usable in the early undergraduate years and to both mathematics majors and others, matrix concepts are emphasised initially as being somewhat less abstract. However, linear transformations are introduced early in the text, and a later chapter illustrates how both viewpoints can be utilised in establishing various theorems.

An attempt is made to motivate many of the theorems and definitions, and most of the theory is well-illustrated with concrete examples. There is a wide variety of problems, both simple and difficult, and answers are provided.

The number of slips and misprints does not seem to be excessive for a new text - perhaps one per ten pages.

Chapter headings are: The Algebra of Matrices, Linear Equations, Vector Spaces, Determinants, Linear Transformations, Eigenvalues and Eigenvectors, Inner Product Spaces, and Applications to Differential Equations. There should be adequate material for a one semester course, or perhaps a two quarters course.

2.1. Review by: Joseph Kist.
Mathematical Reviews MR0241558 (39 #2898).

This monograph is devoted to characterising the lattice of proper congruences on a completely 0-simple semigroup.

3.1. Review by: Nathaniel Chafee.
Quarterly of Applied Mathematics 29 (1) (1971), 164.

This book is a text designed to accompany a one-semester introductory course in linear algebra and the theory of linear ordinary differential equations. The authors have in mind a course to be taken by mathematics and science students at the sophomore or perhaps junior year level. It is assumed that these students have already had two or three semesters of calculus with perhaps a very brief introduction to the theory of ordinary differential equations.

The book is divided into seven chapters. In the first chapter the authors present eight problems from various areas of science and technology. Each of these problems involves either a system of linear algebraic equations in several unknowns or one or more linear ordinary differential equations. While discussing each problem the authors stress the phenomenon of linearity, and they indicate that the given problem will be solved as appropriate techniques are developed in the remaining six chapters of the book.

The second chapter concerns matrices. Addition and multiplication of matrices are defined and discussed. The notion of a non-singular matrix is introduced.

In Chapter 3 the authors take up the theory of systems of linear algebraic equations in several unknowns. The presentation is thorough. Essentially, the student is introduced to the method of Gaussian elimination and some related ideas. However, he is not yet introduced to the concept of a linear space or any of the notions consequent there from.

Chapter 4 concerns determinants. Here it is shown that an $n \times n$ matrix is non-singular if and only if its determinant is non-zero. The chapter also includes Cramer's Rule.

In Chapter 5 the student meets the concept of a linear space, and he learns about the notions of linear independence and dimension. These ideas are then used to develop further the theory presented in Chapter 3. The student learns the relation between the homogeneous and non-homogeneous linear equation in Euclidean $n$-space.

In Chapter 6 the authors present the theory of linear ordinary differential equations. They discuss the notions of a fundamental matrix solution and a Wronskian. They discuss the homogeneous equation with constant coefficients. The notions of eigenvalue and eigenvector for an $n \times n$ matrix are introduced. (The Jordan canonical form is discussed in an appendix.) They discuss the asymptotic behaviour of solutions as $t \rightarrow ±∞$ particularly for systems in the plane.

Chapter 7 concerns the Laplace transform and its application to linear ordinary differential equations.

The presentation throughout the entire text is thorough, perhaps even painstaking. There are many examples and exercises, some of which are used to introduce new concepts.
...

To conclude, I think that this book will be a useful text for the type of course for which it is intended. On the whole, I believe that the authors have achieved their stated goals in a better than satisfactory manner.

3.2. Review by: Warren S Loud.
American Mathematical Monthly 80 (4) (1973), 451-453.

As the mathematics curriculum has evolved, it has been found desirable to introduce linear algebra at an earlier stage than formerly. This is particularly true for students with interest in science and engineering. One of the important ways in which linear mathematics appears for scientists and engineers is in linear differential equations. Linear differential equations provide a well-motivated blend of finite-dimensional and infinite-dimensional linear spaces, which students can learn with appreciation at an early stage.

The books under review is designed as a textbook for the kind of audience described above. The reader is not assumed to have extensive background in linear algebra or in differential equations. The book gives a fairly extensive introductory treatment in both areas. Restraint is shown in the choice of topics in linear algebra, both to focus on those parts of linear algebra needed for differential equations, and to leave time for. adequate treatment of differential equations in a one-term course.

The book can be well recommended. ... It has eight chapters. The first is a brief introduction showing a number of physical problems related to linear algebra and linear differential equations. Chapters 2, 3, and 4 are entitled Matrix Operations, Linear Systems of Algebraic Equations, and Determinants. These are a basic introduction to linear algebra involving mostly manipulative techniques. The reduction of a matrix to row-echelon form is a primary tool. The readers should have no difficulty with this material. Chapter 5 is entitled Vector Spaces. Euclidean $n$-space is introduced followed by abstract vector spaces, subspaces, span, linear independence, basis, and linear transformations. The pace is rapid and readers may find the material difficult. Chapter 6 is entitled Linear Systems of Differential Equations. Linear vector systems are introduced at once, with the scalar equation case interwoven. Existence and uniqueness for initial value problems is stated, with the proof in an appendix. Linear homogeneous systems, solution matrices, fundamental matrices are introduced and the linear algebraic properties of the solutions established. The linear equation with constant coefficients is discussed with emphasis on the second-order case. For the nonhomogeneous case, there is a return to vector systems. The variation of parameters formula is given. The method of undetermined coefficients for nonhomogeneous equations with constant coefficients is not discussed. The pace is fairly rapid, and students with no background in differential equations might find vector systems hard to understand at once. Chapter 7 is entitled Eigenvalues, Eigenvectors, and Linear Systems of Differential Equations with Constant Coefficients.

The exponential of a matrix is defined. Eigenvalues and eigenvectors of a matrix are defined. The matrix $exp (At)$ is shown to be a fundamental matrix of the system $x' = Ax$ and various methods for its calculation are given. The case of multiple eigenvalues for $n$th order systems is deferred. Real homogeneous systems of order two are discussed with the various kinds of critical points studied. Chapter 8 is on the Laplace Transform with enough development for treatment of linear equations and linear systems with constant coefficients. It is probably not possible to get to this chapter in a one-term course. There are three appendices, one on exponentials of matrices, one on a proof of the existence theorem, and one leading to Jordan Canonical Form.

Brauer, Nohel, and Schneider are concrete in their introduction to linear algebra. They allow the student to have manipulative experience with matrices, linear equations, and determinants before abstract linear spaces are introduced. The abstract situation is then treated quite rapidly. ... They (perhaps unfortunately) definitely assume a half-term background in differential equations. As a consequence a student with no background in differential equations will find the pace rapid and will have to learn the elementary techniques indirectly from the first chapter or from other sources.

4.1. Review by: Editors.
Mathematical Reviews MR0349698 (50 #2191).

The main difference between this and the first edition [1968] is the addition of a new section to illustrate the geometric content of Sylvester's theorem. The book is intended as a first course on the subject and covers the standard material, going as far as a statement of the theorem on the Jordan canonical form, a substantial chapter on inner product spaces, and simple applications to systems of linear differential equations.

4.2. Review by: John Asquith.
The Mathematical Gazette 58 (405) (1974), 239.

Another book on elementary linear algebra prompts one to look at any new offering with a view to novelty of presentation and clarity of exposition. This book was written mainly for American undergraduates other than mathematical specialists. After a short introduction to matrices, there follows a chapter on the solution of linear equations. This prepares the ground necessary for the study of abstract vector spaces. Chapters follow on determinants, linear transformations, eigenvalues and similarity with some rapid switching between the matrix and linear transformation points of view. Two chapters on inner product spaces and systems of linear differential equations complete the book.

This is a well written introduction to the subject. The exposition is clear and little is assumed in the way of previous knowledge. However, I was surprised at the lack of applications, bearing in mind the intended readership. Also, it is regrettable that the book is so highly priced, and this must be a decisive factor. There are other equally good introductions to the subject available at less than half the price of this book.