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Linear Algebra: A Geometric Approach

Theodore Shifrin and Malcom R. Adams
W. H. Freeman
Publication Date: 
Number of Pages: 
[Reviewed by
Gizem Karaali
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The first reaction from those reading this review will most likely be “yet another linear algebra book! why?” I have to admit that was my reaction, too, when I first got my hands on this book. I was more or less expecting yet another watered-down text, with more uninspiring visuals than useful explanations and more tedious matrix computations than clear theoretical interpretations. The good news is that I was wrong. This is a well-written textbook which focuses on the geometric interpretation of the basic concepts of linear algebra but does not hesitate to go into the abstract notions that make the whole subject stand on its own as a glorious chapter of modern mathematics. I am still not sure if I will drop my own favorite text for this one for my next linear algebra course, but it certainly presents a good alternative to the many books out there.

The basic premise is the familiar one that linear algebra should be taught with geometry in mind. That linear equations correspond to linear spaces and their simultaneous solutions can be viewed most profitably via a geometric approach is nothing new to most readers of MAA Reviews. However sometimes in our rush we may forget to incorporate this basic idea into our courses. Depending on the choices we make, linear algebra can become either the most exciting mathematics course in the lower division or into a tedious mix of matrix calculations and definitions-theorems-proofs which are unmotivated, misunderstood and unappreciated. Hopefully this review will help you decide whether this book is going to be included in your choices for your next linear algebra course.

Shifrin and Adams start with vectors on the plane and dot products, and move on to n dimensions (including a discussion of hyperplanes in Rn) after which they introduce the basic ideas of linear systems. The geometric connection is there from the beginning, and the first parts of this chapter have about as many figures as pages. Matrix algebra is studied in the second chapter, with the basic matrix operations, matrix inverses, and the transpose each getting their own subsections. The third chapter introduces vector spaces. First the focus is on subspaces of Rn and the basic notions like linear independence, basis and dimension are all studied within this more concrete setting. Four basic subspaces associated with a matrix (the nullspace, the row and column spaces and the nullspace of the transpose) are studied in detail. An optional section on abstract vector spaces concludes this chapter.

The natural progression to the study of linear transformations follows in the subsequent chapters. Projections and changes of bases are studied in Chapter 4 along with the development of other fundamental concepts like inconsistent systems and orthogonality. Determinants first show up, in Chapter 5, as signed areas; the later generalizations to higher dimensions make perfect sense with this motivating example in mind. Chapter 6 is on eigenstuff; it begins with the characteristic polynomial and wraps things up with the spectral theorem. A seventh chapter presents a few further topics like the Jordan canonical form and applications to computer graphics and differential equations.

The text makes a serious effort to embed the basic notions of mathematical proof into the main flow. The authors intend it to be used for a course introducing the basics of linear algebra while also preparing the students for more advanced mathematics courses where they will be reading and writing proofs of their own. This makes the text more appropriate for courses which are transitional in nature, where the audience includes students who are looking to become mathematics majors, rather than for courses where the sole purpose is to introduce the main tools of liner algebra to future physicists, engineers and economists. The informal language of the text is interrupted often with more precisely stated definitions and theorems, and the students are gradually guided into thinking more rigorously and provided with progressively sophisticated exercises to test their developing skills in writing proofs. The instruction on writing proofs is not found in one separate section or in an appendix. Instead many blue boxes are sprinkled throughout the text, where various methods of proof are introduced and hints are given about how to attack a particular kind of problem (eg. asserting set equality, or showing linear independence of a collection of vectors). The almost seamless way the informal and the formal are combined in the book make the book feel like a well-polished set of lecture notes, but in a good way.

The brief description I gave above probably makes it pretty obvious that the Shifrin-Adams book does not attempt to revolutionize the teaching of linear algebra. In fact the table of contents is pretty traditional. The emphasis on geometry is also not incredibly novel; there are many other texts which focus on visuals and concrete geometric analogies to motivate students (I reviewed one such book for MAA Reviews: Visual Linear Algebra). The notion that linear algebra can be used as a suitable context for introducing students to the rigors of upper level mathematics is also not really unorthodox, as many colleges and universities are already using this idea. It is mainly the successful combination of all these features that makes this book interesting and worthy of looking seriously into.

A final comment: I don't know if it might be considered cheating, but I tend to check out other reviews of a book before wrapping up my own. In this case I visited the page for this book in and the slew of negative comments from students was, for me, a wake-up call. It is not uncommon that students have completely unexpected experiences with a text no matter how scrupulous the instructor may have been in her search for the best textbook to use. Of course instructors make choices with many concerns in mind, including pedagogy, pricing, examples and exercises, but sometimes we end up making what turn out to be unpopular choices. Being open to student feedback and processing it thoughtfully may lead to a change in course book adoptions, or alternatively may motivate us to make other modifications in our classroom presentations incorporating the text into the course in novel and interesting ways. One of my long-time favorite texts in linear algebra was slammed by my first class, but has become a treasured reference (if not a smashing hit with) for the following ones.

Gizem Karaali is assistant professor of mathematics at Pomona College.

1. Vectors and Matrices
    1. Vectors
    2. Dot Product
    3. Hyperplanes in Rn
    4. Systems of Linear Equations and Gaussian Elimination
    5. The Theory of Linear Systems
    6. Some Applications
2. Matrix Algebra
    1. Matrix Operations
    2. Inverse Matrices
    3. The Transpose
3. Vector Spaces
    1. Subspaces of Rn
    2. Linear Independence
    3. Basics and Dimension
    4. The Four Fundamental Subspaces
    5. A Graphic Example
    6. Abstract Vector Spaces
4. Projections and Linear Transformations
    1. Inconsistent Systems and Projection
    2. Orthogonal Bases
    3. Linear Transformations
    4. Change of Basis
5. Determinants
    1. Signed Area in R2
    2. Determinants
    3. Cofactors and Cramer's Rule
6. Eigenvalues and Eigenvectors
    1. The Characteristic Polynomial
    2. Diagonalizability
    3. Applications
    4. Spectral Theorem
7. Further Applications
    1. Complex Eigenvalues and Jordan Canonical Form
    2. Computer Graphics and Geometry
    3. Matrix Exponentials and Differential Equations