Linear Algebra by Messer
The full title is Robert Messer, Linear Algebra: Gateway to Mathematics, ISBN-10: 0065017285 | ISBN-13: 978-0065017281
I used the book several times to teach a linear algebra course. It is well written and contains plenty of good exercises of various levels.
In chapter 1, you are thrown at the definition of vector space without preparation. The definition is a long list of axioms and it's tough on the student. What follows is also tough but inevitable -- using the axioms to prove some "simple" facts about the algebra of vectors. Discussing Euclidean spaces would provide some context for the student. I was glad to see quotient spaces and vector fields as optional topics.
In chapter 2, systems of linear equations are introduced. There is, again, no context. Meanwhile, spending so much time on elementary row operations seems wasteful.
In chapter 3, the standard topics of linear combinations, linear independence, span, basis, and dimensions are discussed. Some proofs here are too terse in my opinion. I was good to see infinite dimensional spaces as an optional topic. Many good examples are given.
In chapter 4, inner product spaces are introduced axiomatically. I'd prefer to put this topic at the end of the course.
In chapter 5, 6, and 7, one goes to matrices, then to linear operators, back to matrices, and then the determinants. I was pleased to see a few commutative diagrams. Dual spaces are included as an option.
The rationale for the second part of the title is that it can serve as an introduction to pure mathematics: definitions, proofs, etc. This would be tricky and there are better ways to do it. I wish I saw more connections to calculus. The book is a bit pricey.