Linear Algebra by Messer
From Intelligent Perception
The full title is Robert Messer, Linear Algebra: Gateway to Mathematics, ISBN-10: 0065017285 | ISBN-13: 978-0065017281
The rationale for the second part of the title is that it can serve as an introduction to pure mathematics, proofs etc. That part is tricky and I am not sure this is a good way to start. But the book is good as a first course in linear algebra.
Well written. Good exercises and plenty of them. I wish I saw more connections to calculus.
Used it twice for Linear algebra: course.
A bit pricey.
Chapter 1. Vector spaces.
You are thrown at the definition of vector space in Section 1.2 without preparation. That's tough. What follows is also tough but inevitable -- using the axioms to prove some "simple" facts about the algebra of vectors. Still, this part is a bit too long.
"Closed" under operations has been a hard topic for the students. You've got to push infinite dimensional spaces / function spaces to make them confront the issue, and the need to use axioms and theorems, less intuition.
More on Euclidean spaces is necessary in this context.
Chapter 2. Systems of linear equations.
Need some examples here! Where do they come from? Why do we need these?
Too much on elementary row operations.
Numerical methods of linear algebra, important but no time.
Chapter 3. Dimension theory.
Some proofs here are "compact" for better or worse.
I didn't use $[v]_B$ for the coordinates of $v$ with respect to basis $B$, too cumbersome.
Infinite dimensional spaces only as an optional topic. Good examples.
Chapter 4. Inner product spaces.
I put this at the end of the course.
More motivation is needed for the new structure.
Continuity is optional.
Chapter 5. Matrices.
There is a motivational example here.
I skipped Markov chains.
Chapter 6. Linearity.
I didn't use $[f]_B$ for the matrix of $f$ with respect to basis $B$, too cumbersome.
Glad to see commutative diagrams but I did them differently.
Glad to see dual spaces as an option.
Chapter 7. Determinants.
More "compact" proofs here...
I skipped the cross product.
Chapter 8. Eigenvalues and eigenvectors.
Eigenvalues have to be real?