It covers most of the topics covered in a first course in linear algebra, it’s just very application-specific. It has some basic proofs in the exercises, but nothing overly difficult, and the more involved proofs give you explicit instructions on the steps to take.
There is a chapter on abstract vector spaces and there are a few examples given besides the usual R^n (polynomials, sequences, functions) but there is almost no time spent on these. There is also no mention of the fact that the scalars of a vector space need not be real numbers; that you can define vector spaces over any number field.
There is only a passing discussion of complex numbers (as possible Eigenvalues and in an appendix) but no mention of the fact that vector spaces over the field of complex numbers exist and have an even more well-developed theory than for real numbers.
But more fundamental than a laundry list of missing or unnecessary topics is the fact that it’s application focused. Pure mathematics courses are proof and theory focused. So they cover all the same (and more) topics in much richer theoretical detail, and they teach you how to prove statements. Pure math students don’t just learn how to write proofs in one or two courses and then move on; all of the courses they take are heavily proof-based. Writing proofs, like programming, is a muscle that benefits from continued exercise.
So if you’re studying (or previously studied) science or engineering and learned all your math from that track, switching to pure math involves a bunch of catch up. I’ve met plenty of people who successfully made the switch, but it took a concerted effort.
There seems to be a fundamental difference in mindset between the “applications” based learning of mathematics, and this pure math based version. Are there benefits to be had for a person that only intends to use mathematics in an applied fashion?
This depends on who you ask. Personally, I found studying pure math incredibly rewarding. It gave me the confidence to be able to say that I can look at any piece of mathematics and figure it out if I take the time to do so.
I can't speak for those who have only studied math at an applied level directly. My impression of them (as an outsider but also a math tutor) is that they are fairly comfortable with the math they have been using for a while but always find new mathematics daunting.
I have heard famous mathematicians describe this phenomenon as "mathematical maturity" but I don't know if this has been studied as a real social/educational phenomenon.
There is a chapter on abstract vector spaces and there are a few examples given besides the usual R^n (polynomials, sequences, functions) but there is almost no time spent on these. There is also no mention of the fact that the scalars of a vector space need not be real numbers; that you can define vector spaces over any number field.
There is only a passing discussion of complex numbers (as possible Eigenvalues and in an appendix) but no mention of the fact that vector spaces over the field of complex numbers exist and have an even more well-developed theory than for real numbers.
But more fundamental than a laundry list of missing or unnecessary topics is the fact that it’s application focused. Pure mathematics courses are proof and theory focused. So they cover all the same (and more) topics in much richer theoretical detail, and they teach you how to prove statements. Pure math students don’t just learn how to write proofs in one or two courses and then move on; all of the courses they take are heavily proof-based. Writing proofs, like programming, is a muscle that benefits from continued exercise.
So if you’re studying (or previously studied) science or engineering and learned all your math from that track, switching to pure math involves a bunch of catch up. I’ve met plenty of people who successfully made the switch, but it took a concerted effort.