And precisely this is the risk of teaching "mathematics" using a programming language. Unfortunately, I think quite a few people don't actually know the abstract mathematical definitions and use concrete implementations as definitions in their heads in the first place.
At least using a theorem prover would get you closer to doing actual maths and proving.
It's probably more accurate to say courses like these teach mechanics/calculation more than they teach the theory.
There are already an uncountable number of linear algebra textbooks on the market. This is not meant to be a formal math textbook, but is instead aimed at those looking to pick up practical application-level insights and skills. (The "Why this matters" sections in the book are a worthy improvement on the state of the learning art by themselves, IMHO.)
This collection of lab exercises seems far better suited to the purpose than the abstract proof-based resources that some of you seem to have in mind. Fortunately, those resources are still available to those who want to go further into the theory.
And if you feel you have something to add to linear-algebra pedagogy yourself, well... what's stopping you?