Looking for a different type of Linear Algebra book

I might have just the thing: a book called Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra by Jiri Matousek (information here: https://bookstore.ams.org/stml-53), which covers some 'well-known mathematical gems' as well as some lesser-known results, mostly in combinatorics and graph theory, and requiring only a modest amount of linear algebra as a prerequisite.

The book Linear Algebra in Action by Harry Dym is also worth a look. After the first dozen or so chapters covering the standard material, he outlines many applications in other (mostly analysis-related) areas of math, such as difference and differential equations, extremal problems and convex analysis, dynamical systems, Markov chains, and even some complex analysis!

As for LA books that cover computation of homology groups, for instance, I'm guessing these are rather rare, as this topic requires so much background material that expecting it to be covered in a linear algebra book (introductory or otherwise) is unrealistic. Also, if you're interested in graph theory applications in particular, just pick up any book on 'spectral graph theory', a field that analyses graphs by representing them with matrices and studying their spectra (i.e., sets of eigenvalues), and is unsurprisingly very linear-algebraic.

Lastly, and for the record, I do actually find the exercises in Axler to be super-exciting!

Here are a couple I found as well.

Computational Linear & Commutative Algebra looks like there are a lot of heavily algebraic-geometric flavored applications of linear algebra. And actually, this is the 3rd in the series, the other two books are just called Computational Commutative Algebra 1/2, but I'm guessing there is quite a bit of linear algebra in use too, and they have a lot of "tutorials" where they cover e.g. polytopes, graph colorings, varieties, chess puzzles, error correcting codes, and much more.

And, there's no homology theory here, but Artin's Algeba of course makes heavy use of linear algebra and matrix groups throughout the text with plenty of fun, challenging exercises.

Neither one, of course, is a replacement for a first class in theoretical linear algebra (actually, maybe Artin could be used this way... maybe), but I think that's ok.