Pedagogically, the challenge to teaching linear algebra is that you start with "here's systems of linear equations, we can put them into matrices and now here's row operations to solve them," and you end up with "now matrices are actually representations of linear operators on vector spaces, let's analyze the properties of this specific operator." Usually, this is also coupled with a reluctance to actually discuss vector spaces, since the meat of it involves abstract algebra, which usually comes after linear algebra.

Failing to tackle this challenge appropriately can leave students confused about properties that seem apparently random (trace and determinant are big offenders here), or textbooks bringing something up only to never mention it again (null space is often an example here). On top of this, there is also the multiple notation problem (admittedly, not as bad as calculus, where there are too many notations for derivative) and the minor issue that many of the algorithms taught in the book aren't used in practice because of numerical stability issues.

It has been so long since I've taken linear algebra, and I've taken abstract algebra courses since then, that I can't really compare this book to the approach that I learned. Skimming the book, the thing that jumps out the most to me is that LU factorization and determinants are shoved surprisingly late in the book [1], and eigenvalues are "previewed" quite early. I'm not sure that's a good approach: LU factorization is important because backsolving the L and U matrices is more numerically stable (and sparser, when you're dealing with sparse matrices) than the inverse matrix, and it works even if your matrix isn't square. Furthermore, determinants tie in better to row operations, and their weird application with Cramer's rule is another way to solve a set of linear equations: you don't want to introduce Cramer's rule months after you finished treating matrices as stepping stones to solving linear equations.

The book does cover vector spaces, although in a bit of a dance around not covering abstract algebra. I'm not sure it's an effective introduction of vector spaces, although it could well suffice to ease the pedagogical trap mentioned earlier. On the other hand, if it's going to dive that far into vector spaces, it would probably be helpful to have some more sections on matrices over fields that aren't real numbers (i.e., complex numbers (make sure to mention conjugate transpose and Hermitian matrices!), rational numbers, and finite fields).

[1] Strassen's algorithm for matrix multiplication is described before LU factorization, to give you an idea of how weird the ordering ends up being.