[Koshkin]

Well, perhaps it's not meant to be the first book on mathematics one sees in their life.

[margalabargala]

From the second paragraph of the introduction to the book we are discussing:

> 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 it's certainly meant to be the first math book one sees in their life that discusses rigorous proofs.

[echion]

It's supposed to be the introduction to rigorous proof, so it is supposed to be "the first book on [formal] mathematics" for its audience.