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.

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.