Linear Algebra (Hefferon)

jim-hefferon‘s undergraduate linear algebra textbook. Fourth edition, 2020; online since 1996. Dual-licensed under the GNU Free Documentation License or CC BY-SA 3.0 US, at the reader’s choice, with the LaTeX source on GitLab — free in the copyleft sense, the same footing as corral-vector-calculus.

This closes the spoke’s largest hole: linear algebra sat under every text in the corpus and was the subject of none (synthesis open question 4).

What it covers

The standard one-semester arc, in five chapters:

1. Linear systems — Gauss’s method, reduced echelon form.
2. Vector spaces — subspaces, spanning sets, linear independence, basis and dimension.
3. Linear maps and matrices — homomorphisms, isomorphism, matrix representation, change of basis, projection.
4. Determinants — definition, geometry of determinants, Laplace’s expansion.
5. Eigenvectors and eigenvalues — similarity, diagonalization, Jordan form.

Each chapter closes with supplemental Topics — self-contained sections meant for reading, projects or group work rather than lecture, which is where the applications live.

The pedagogical position

Hefferon proves every result, and that is the point of contrast with the rest of this corpus. What he declines to assume is that the student already handles abstraction: the book gets there through motivation, worked computational examples and heavy exercise sets, rather than by starting abstract and staying there. The prerequisite is one semester of calculus — one course, and not the multivariable one.

That combination — full rigour, low entry — is a third position on the axis this spoke has been tracking. Corral rates his own rigour 5/10 and says so in the preface; McMullen is terse notes that assume a 1950 textbook. Hefferon gives up neither proof nor accessibility, and pays for it in length.

Why T1

The corpus’s grading has been contested here before (see synthesis → Provenance versus usefulness), so state the case. Hefferon is a professor of mathematics at Vermont; the book took the Mathematical Association of America’s 2020 Daniel Solow Author’s Award for impact on undergraduate mathematics education, citing its “clear writing style, tremendous variety of exercises”; it is in use in hundreds of university courses. A professional society’s award is peer judgement in the sense the tier ladder means, which is what separates this from Corral’s self-published T2 — not the licence, the production stack, or the free price, all of which the two share.

Supplements

A complete solutions manual with worked answers to every exercise, beamer slides, a Sage lab manual, and video lectures. The Sage manual bears directly on this spoke’s pedagogy thread: it names the computational tool the course assumes, where Corral assumes none.

Connections

• author jim-hefferon
• fills the hole named in synthesis and under mathematics
• free-licence sibling of corral-vector-calculus; the two make the licence/rigour axes independent
• linear algebra is the machinery under vector-calculus and fourier-analysis