FWiW I started out in Engineering and transferred out to a more serious mathematics | physics stream.
The Engineering curriculum as I found it was essentially rote for the first two years.
It had more exams and units than any other courses (including Medicine and Law which tagged in pretty close) and included Chemistry 110 (for Engineers) in the Chemistry Department, Physics 110 (for Engineers) in the Physics Department, Mathematics 110 (for Engineers) in the Mathematics Department, and Tech Drawing, Statics & Dynamics, Electrical Fundementals, etc in the Engineering Department.
All these 110 courses for Engineers covered "the things you need to know to practically use this infomation" .. how to use Linear Algebra to solve loading equations in truss configurations, etc.
These were harder than the 115 and 130 courses that were "{Chemistry | Math | Physics} for Business Majors" etc. that essentially taught familiarity with subjects so you could talk with the Engineers you employed, etc.
But none of the 110 courses got into the meat of their subjects in the same way as the 100 courses, these were taught to instruct people who intended to really master. Maths, Physics, or Chemistry.
Within a week or two of starting first year university I transfered out of the Maths 110 Engineering unit and into Math 100, ditto Chem and Physics. Halfway through second year I formally left Engineering the curriculum altogether (although I later became a professional Engineer .. go figure).
The big distinction between Math 100 V. Math 110 was the 110 course didn't go into how anything "worked", it was entirely about how to use various math concepts to solve specific problems.
Math 100 was fundementals, fundementals, fundementals - how to prove various results, how to derive new versions of old things, etc.
Six months into Math 100 nothing had been taught that could be directly used to solve problems already covered in Math 110.
Six months and one week into Math 100 and suddenly you could derive for yourself from first principals everything required to be memorised in Math 110 and Math 210 (Second year "mathematics for engineers").
I'm incredulous that a linear algebra course taught by mathematics faculty didn't have a lot of theorem proving.
Maybe that would be the case if the intended audience is engineering students. But for mathematics students, it would literally be setting them up for failure; a student that can't handle or haven't seen much theorem-proving in linear algebra is not going to go very far in coursework elsewhere. Theorem proving is an integral part of mathematics, in stretching and expanding tools and concepts for your own use.
Maybe the courses are structured so that mathematics students normally go on to take a different course. In that case, GP's point would still have been valid - the LA courses you took were indeed ones planned for engineering, not for those pursuing mathematics degrees. At my alma mater, it was indeed the case that physics students and engineering students were exposed to a different set of course material for foundational courses like linear algebra and complex analysis.
Just like compiler theory, if you don't write compilers maybe it's not that useful and you shouldn't be spending too much time on it, but it would be presumptuous to say that delivering a full compiler course is a fundamentally incorrect approach, because somebody has to make that sausage.
I can only speak to my own experiences, but the math courses were not customised for engineering students. I sat next to students who were planning to become mathematicians. Linear Algebra was an optional course for me.
Having said that, I’m sure theorem proving was part of it (this was many years ago), I just don’t recall it as being fundamental in any sense. I’m sure that has something more to do with the student than the course work. I liked (and like), maths, but I was there to build my tool chest. A different student, with a different emphasis, would have gotten different things out of the course.
But I think my viewpoint is prevalent in engineering, even from engineers who started with a math degree. The emphasis on “what can I do with this”, relegates theorem proving to annoying busywork.
I can second this, in my Engineering degree the Linear Algebra course (and the Calculus course) were both taught by the Math Faculty at my Uni.
The textbook we used was "Linear Algebra: And its Applications" by David C Lay 20 years later I still keep this textbook with me at my desk and consult it a few times a year when I need to jog my memory on something. I consider it to be a very good textbook even if it doesn't it doesn't contain any rigorous proofs or axioms...
Engineers can learn linear algebra from an engineering perspective, i.e. not emphasizing proofs, and that’s fine, but the books being discussed are not intended for that audience.
