Most university math curriculums have a clear demarcation between the early computation-oriented classes (calculus, some diff eq.) and later proof-oriented classes. Traditionally, either linear algebra or abstract algebra is used as the first proof-oriented course, but making that transition to proof-based math at the same time as digesting a lot of new subject matter can be brutal, so many schools now have a dedicated transition course (often covering a fair bit of discrete mathematics). But there's still demand for textbooks for a linear algebra course that can serve double-duty of teaching engineering students a bag of tricks and give math students a reasonably thorough treatment of the subject.
> this book is meant for the audience who can read and write proofs
It seems like the opposite is true:
"It is intended for a student who, while not yet very familiar with abstract reasoning, is willing to study more [than a] "cookbook style" calculus type course."
(from the link).
If your point is one can't learn linear algebra before learning "abstract [mathematical] reasoning"...don't think you're the main target audience of a subject as practical as linear algebra.
> Besides being a first course in linear algebra it is also supposed to be a first course introducing a student to rigorous proof, formal definitions---in short, to the style of modern theoretical (abstract) mathematics.
So I think it's fair to say that the book (ought to) assume zero knowledge of proofs, contra your parent's claim that the audience is expected to be able to read and write proofs.
> Syntax rigour has almost nothing to do with correctness.
I see your point: has almost nothing correctness with rigour do to Syntax.
Syntax rigor has to do with correctness to the extent that "correctness" exists outside the mind of the creator. Einstein notation is a decent example: the rigor is inherent in the definition of the syntax, but to a novice, it is entirely under-specified and can't be said to be "correct" without its definition being incorporated already...which is the ultimate-parent-posts' point and I think the context in which the post-to-which-you're-replying needs to be taken.
And if you're going to argue "This is just wrong?" (I love the passive-aggressive '?'?) while ignoring the context of the discussion...QED.