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Linear Algebra: From the Beginning

Eric A. Carlen and Maria C. Carvalho
W.W. Freeman
Publication Date: 
Number of Pages: 
[Reviewed by
Tom Schulte
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Aptly subtitled “From the Beginning,” this text is a self-contained book that works both as a first-semester text for linear algebra and as a guided introduction for those who desire on their own to gain entry into the land of linear algebra with no more of a passport than a working concept of vectors. The additional phrase on the cover, “for scientists and engineers,” resonates with the roll-up-your-sleeves approach that dives into practical mechanics, but the book is largely devoid of applications examples beyond basic curve-fitting, digital signal processing, and a few other things. While the exercises are present to keep the reader on track, someone making the voyage alone will find that the online solutions manual is no longer available, but may appreciate the new version professionally edited and published by W.H. Freeman and Co. (ISBN-10: 1-4292-0428-1).

Based upon their working experience with undergraduate students, the authors seek to teach linear algebra in a way that minimizes the required mathematical sophistication. This makes the text good for readers that want to learn the subject on their own. Introductions to mathematical reasoning are made by bringing in geometric perspectives, building up higher-dimensional theorems from 2-dimensional ones, etc. However, this is an introductory text, in no way dense with pure mathematics and theory. As a study aid, the book prefaces the beginning of the chapters with a listing of “very important formulas” covering matrix basics referred to throughout the text.

In sections packed with plenty of examples and exercises the authors describe vectors, matrices and linear transformations, the solution set of a linear system, the image of a line translation, determinants, the Eigenvalue problem and Eigenvectors, and abstract vector spaces. Although intended for classroom use, this could also serve as a self-study text on practical approaches to solving well-defined equations and to elucidate, within the problem-solving context, the mysteries of the dot product in higher dimensions, the discrete Fourier transform and other approaches to diagonalization, and the Jordan canonical form.

Tom Schulte is working on a PhD in Applied Mathematics at Oakland University and this year checked the box for the SIAM Activity Group on Linear Algebra when renewing his SIAM membership.

1.Vectors, Matrices, and Linear Transformations
2. The Solution Set of a Linear System
3. The Image of Linear Transformation
4. Determinants
5. The Eigenvalue Problem
6. More About Eigenvectors and Eigenvalues
7. Abstract Vector Spaces