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Linear Algebra for the 21st Century

Anthony Roberts
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Peter Olszewski
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The singular value decomposition (SVD) is the central computation method used in Linear Algebra for the 21st Century.  The author, Anthony Roberts, makes the case that the SVD is needed in such applications of data mining, AI, and in bioinformatics.  The texts sets the stage for professors to have a modern treatment of Linear Algebra by using the SVD throughout the text with no loss of rigor.  In the early parts of the text, there is a great deal of emphasis places on orthogonality.  There are also many graphical pictures and problems presented, which will help students visualize solutions.
The text is split into seven chapters: Vectors, Systems of Linear Equations, Matrices Encode System Interactions, Eigenvalues and Eigenvectors of Symmetric Matrices, Approximate Matrices, Determinants Distinguish Matrices, and Eigenvalues and Eigenvectors in General.  Before opening up Chapter 1, my one opening comment is to have Chapters 2 and 3 switched: Introduce matrices and their operations first before discussing systems of linear equations.  This maybe a personal preference for certain professors so, I digress.  Chapter 1 gives the student a review of Vectors.  Several examples and proofs are presented along with some applications that can be used with MATLAB or Octave.  
On page 193, Theorem 3.3.6 presents the key backdrop of SVD Factorization.  My only two comments is that the author should use quotes around the word diagonal in the third bullet since the matrix S need not be square, as shown in Example 3.3.14 on pages 198-200.  Also, I strongly believe a worked out example on how to find the SVD of the form \( A = USV^{T} \) should be presented, as students will ask in class.  
Throughout the text, there are many motivational quotes.  One in particular I like is found on page 210: “When doing maths there’s this great feeling.  You start with a problem that just mystifies you.  You can’t understand it, it’s so complicated, you just can’t make head nor tail of it.  But then when you finally resolve it, you have this incredible feeling of how beautiful it is, how it all find together so elegantly,” (A. Wiles, 1993).  This is especially true with Example 4.2.22 on pages 400-401 where the student must identify the conic section given the equation \( x^{2} + 3xy -3 y^{2}- 1/2=0 \) by rotating the coordinate system.  Also, on pages 550-552, Example 7.1.22 on orangutans, presents a very detailed application solution using a model to predict the population for the next five years.
While there are texts that devote some sections that discuss SVD as in Lay and Strang, this is the first text I have read that uses SVD as the main operation to solve systems of linear equations instead of the traditional augmented matrix and elementary row operations to obtain a reduced row echelon form.  While this book has been published in 2020, it is hard to judge how students and professors will receive this book.  A nice addition to the text could be an online resource of practice problems for COVID-19 remote learning.  I can see mixed reviews of the text as one always needs to have or find the factorization of a matrix.  I’m certainly willing to try using this text the next time I’m assigned Linear Algebra and report back on my findings. 
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 Webpage: