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Axler does limit itself to vector spaces over real and complex fields, though. That‘s fine, but I would have appreciated notices, which proofs and theorems do
by Tomte 2mo ago
Axler does limit itself to vector spaces over real and complex fields, though.
That‘s fine, but I would have appreciated notices, which proofs and theorems do not hold in the general case.
It‘s an exercise for the reader.
- fn-mote 2mo agoThis is basically a request for a graduate course, though. Definitely not a first course in linear algebra. TBH, I don’t think there are surprises in the “general case” (whatever that is… modules over a PID??) that you can’t see by understanding the real&complex situation.
- Tomte 2mo agoNo, but a second course. Still firmly in undergraduate territory, second semester at my university. Most undergraduate linear algebra textbooks also contain this, unless they are matrix computation heavy (usually for the first course).
- seanhunter 2mo agoThat’s not true at all. He does precisely the thing you want him to do and also gives a pedagogical justification in the preface. > This book usually develops linear algebra simultaneously for real and complex vector spaces by letting F denote either the real or the complex numbers. If you and your students prefer to think of F as an arbitrary field, then see the comments at the end of Section 1A. I prefer avoiding arbitrary fields at this level because they introduce extra abstraction without leading to any new linear algebra And the remarks at the end of 1A are that if you want to, you can think of F as an arbitrary field everywhere except the sections on inner product spaces and where the given field is C you can frequently also use any other algebraically closed field.