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> if you're not comfortable with the basics of mathematical proofs, then you're not ready for the subject of linear algebra

That's quite far from what the author of the book you're discussing wrote:

> [this book/course] is also supposed to be a first course introducing a student to rigorous proof, formal definitions

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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.

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