nculwell.github.io

Linear Algebra

Introductions

Applied focus

• Strang. Introduction to Linear Algebra (with OCW lectures) (4e)

  The latest incarnation of Gilbert Strang’s linear algebra textbook. I haven’t seen this one myself, but it goes with his highly-regarded online LA course. It is reportedly somewhat more elementary than his older Linear Algebra and its Applications.

• Strang. Linear Algebra and its Applications (4e, 4e intl, 3e)

  Strang’s older book.

• Hefferon. (FREE ONLINE, Dec 2014 paperback)

  Jim Hefferon wrote this free (GFDL/CC BY-SA 2.5) linear algebra text [http://joshua.smcvt.edu/linearalgebra/] which is available as a PDF file or as Latex source. It has exercises with worked answers.

• Klein. Coding the Matrix (With Brown/Coursera course; great course but not much use as a stand-alone text. Also, it’s not required for the course.)

• Lay. Linear Algebra and Its Applications (3e upd, 4e, 5e)

• Poole. Linear Algebra: A Modern Introduction (3e)

  A colorful “modern textbook” approach to linear algebra. Don’t bother with the 4th Edition, as it’s very expensive and largely the same as the Third Edition.

• Singh. Linear Algebra: Step by Step (1e)

  A recent book. Looks like it has a lot of good diagrams (which are often lacking in older textbooks).

• Meyer. Matrix Analysis and Applied Linear Algebra Book and Solutions Manual

  A good book, but expensive. This used to be available online for free, but the author took it down due to unspecified abuses of the license agreement.

• Andreescu. Essential Linear Algebra with Applications: A Problem-Solving Approach (1e)

Abstract focus

These are texts in the old style, aimed at mathematicians. Some of them are older texts that have stood the test of time.

• Hoffman and Kunze. Linear Algebra (2e, 2e intl ed)

  For many years this was the standard LA textbook, though it seems it’s not as popular as it used to be. It has good, thorough coverage of the core of undergraduate linear algebra. It’s a bit old, however, and leans toward the abstract. I don’t think it even mentions the Singular Value Decomposition (SVD).

• Shilov. Linear Algebra (Dover ed)

  A Russian approach to linear algebra. Like many Russian textbooks, it jumps right in and makes no attempt to hold your hand. Even though the content may be appropriate for first-time learners, many students in the US won’t be prepared for a book like this when they start LA. Also tends toward the abstract and doesn’t cover the SVD. Includes hints and solutions for exercises.

• Friedberg, Insel, Spence. Linear Algebra (4e, 4e intl, 3e)

• Lang. Linear Algebra (3e, Solutions)

• Treil. Linear Algebra Done Wrong

  The title is a play on Axler’s Linear Algebra Done Right. Free online at the author’s page: https://www.math.brown.edu/~treil/papers/LADW/LADW.html

• Beezer. A First Course in Linear Algebra (FREE ONLINE)

  Another free (GFDL) linear algebra textbook, by Robert Beezer of University of Puget Sound. There are many experimental aspects in this book, including its style of “numbering” (everything is referred to by alphabetic abbreviations instead of numbers) and its web-oriented organization. I have no idea how this would work as a text but it seems intriguing as a companion to a more traditional text. (If you’re going the free route, Hefferon would probably make a good companion.)

Some calculus books include an introduction to linear algebra. Among these are Apostol’s Calculus and Hubbard and Hubbard’s Vector Calculus. (Most calculus books include some introduction to linear algebra but it’s usually cursory.)

Axler

I put this book in its own category because it’s a bit unique in its appeal. It fills a need that is particular to the current mathematics curriculum. The ordinary intro books these days do not cover enough advanced material for students to comfortably go directly on to the more advanced books listed below. What’s more, many students don’t have enough exposure to reading and writing proofs yet. Axler’s book serves as both a course in abstract linear algebra that covers the essential topics, and a course in proof-reading and proof-writing. However, its contents might be superfluous for someone who started using one of the more thorough introductory books such as Hoffman and Kunze or Shilov.

• Axler. Linear Algebra Done Right (3e, 2e, ERRATA)

  An outstanding introduction to an abstract approach to linear algebra. This book is all about proofs, you won’t find calculations here. It is intended as a second course in linear algebra, to follow a more computation-oriented course, and I agree that this is its best use. You could start LA here, but you’d end up having read a whole book and still not knowing how to solve a linear equation. One of the distinctive things about Axler’s approach is that he introduces the theory of eigenvalues without using determinants (preferring instead to use the fundamental theorem of algebra). Determinants are then introduced in the last chapter. Also, the quality of the Third Edition hardcover is supposedly really nice.

Problems

• Erdman. Exercises and Problems in Linear Algebra http://web.pdx.edu/~erdman/LINALG/Linalg_licensepage.html
• Zhang. Linear Algebra: Challenging Problems for Students (1e)
• Lipschutz. Schaum’s 3,000 Solved Problems in Linear Algebra (1e)
• Lipschutz and Lipson. Schaum’s Outline of Linear Algebra (5e, 4e)
• Halmos. Linear Algebra Problem Book (1e)

Advanced linear algebra (beyond a first course)

Most of these will probably be more accessible to someone who has read one of the above intro books, and Axler’s Linear Algebra Done Right.

• Halmos. Finite-Dimensional Vector Spaces (Martino ed, Benediction ed, Springer hardcover (OOP))

  This is another abstract approach to LA that emphasizes connections to analysis and intends to prepare the student for a course in functional analysis. Interesting problems and a colorful exposition. Martino Fine Books has a good reputation for their physical product, but I don’t know anything about the physical quality of the Benediction Classics edition. (My own copy is the old Springer hardcover.)

• Horn and Johnson. Matrix Analysis (2e, 1e)

• Horn and Johnson. Topics in Matrix Analysis (1e)

• Lax. Linear Algebra and Its Applications (2e)

  This is a grad-level abstract linear algebra text by Peter Lax, not the introductory book by David Lay! The “applications” here are the pure mathematician’s idea of a “application”, which means applying the theory to a problem in mathematics, not applying mathematics to the real world.

• Roman. Advanced Linear Algebra (3e, 3e intl at AbeBooks)

  Another grad-level abstract linear algebra book. In spirit it is similar to Axler’s Linear Algebra Done Right, but from a more advanced perspective.

• Gantmacher. Applications of the Theory of Matrices (Dover ed)

• Loehr. Advanced Linear Algebra (1e)

  Explores linear algebra in conjunction with many topics in algebra and some in analysis.

• Franklin. Matrix Theory (Dover ed)

  A physics-oriented overview of matrix theory, followed by a chapter on physics applications and a chapter on numerical methods.

• Erdman. *Elements of Linear and Multilinear Algebra

  Free notes: http://web.pdx.edu/~erdman/ELMA/multilinear_algebra_pdf.pdf

Others:

• Gentle. Matrix Algebra: Theory, Computations, and Applications in Statistics
• Bernstein. Matrix Mathematics: Theory, Facts, and Formulas
• Macdonald. Linear and Geometric Algebra
• Kaplansky. Linear Algebra and Geometry: A Second Course

Numerical linear algebra

See Numerical Methods in the CS section.

Random matrix theory

• Anderson, Guionnet, Zeitouni. An Introduction to Random Matrices (1e)
• Baik, Deift, Suidan. Combinatorics and Random Matrix Theory (1e)
• Tao. Topics in Random Matrix Theory (1e)