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Linear Algebra for Everyone

Gilbert Strang
Wellesley-Cambridge Press
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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This book (hereafter LAFE) is not drastically different from the author’s standard textbook Introduction To Linear Algebra 5th edition (hereafter ILA), so I will give a brief review comparing them.
Both books are texts for a first course in linear algebra. ILA is the text for MIT’s one-semester 18.06 Linear Algebra course, which covers about the first half of the book (roughly through eigenvectors and Singular Value Decomposition). LAFE is about half the length of ILA and in fact covers about the same topics as the first half of ILA.
Comparing by chapters: Chapters 1–3 of both books cover similar topics and have a similar structure, although somewhat rearranged because of the early emphasis on vector spaces in LAFE; Chapters 4–7 are very similar in both books, although slightly rewritten in LAFE; and after that they diverge.  ILA is twice as long as LAFE, and covers a variety of applications and additional topics in its Chapters 8–12; LAFE has a new chapter 8, “Learning from Data”, on data science. This new chapter is presented as an application of linear algebra, and deals with deep learning, neural nets, and convolutional neural nets. It has a very different nature than the rest of the book and introduces a lot of new concepts.
There are lots of little improvements in the exposition throughout. The author has provided a brief summary of the differences in an online document, New Ideas in Linear Algebra for Everyone.  
The biggest difference in approach is that LAFE introduces vector spaces very early, almost on the first page, and they continue to play a prominent role throughout the book. ILA follows a more traditional and historical path where we start with linear equations, matrices, and solving systems, before 1we move to the more abstract concept of vector space. (In analogy to the many variant calculus texts, one might call LAFE Linear Algebra: Early Vector Spaces.) LAFE introduces the column and row spaces in Chapter 1, along with a new CR (Column-Row) factorization A = CR. The C matrix is made up of columns that are linearly independent and span the column space. The R matrix is constructed to make the factorization work, although it turns out to be the reduced row echelon matrix.
Which book should you choose for a course? It depends on whether you like the “early vector spaces” approach (LAFE), or would rather start with the more traditional emphasis on systems of linear equations (ILA). The books’ list prices are about the same. ILA is twice as long and covers many more topics, although you would not get to the extra topics in a typical one-semester course.

Allen Stenger is a math hobbyist and retired software developer. He was Number Theory Editor of the Missouri Journal of Mathematical Sciences from 2010 through 2021. His personal web site is His mathematical interests are number theory and classical analysis.