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First, by “advanced”, I mean the prerequisites are basic linear algebra (e.g., the course and book by Gilbert Strang) and some linear space theory (e.g., first 8 chapters of Lax’s, but without the more advanced things like dual, quotient space, annihilator, complex space, norm and completeness.)

Second, since I am from a statistics and science background and also interested in abstract math, I hope the book can go back and forth between matrix and linear space. I know some good books in respective trends (Holmes’ for linear space, and Horn’s for matrix theory), or some books dealing linear space and matrix in separated chapters (Lax’s is kind of like this), but none of them cares to take advantages of both trends to enlighten the core results.

Are there any books doing this? Or it is in fact better to study them separately (in two books)? (AFAIK, “linear algebra” and “matrix theory” tend to cover different topics, though I indeed know a book that gives most of results in linear algebra mostly using matrix but not space: Matrix Algebra by Abadir & Magnus.)

I read the most related question here: Book to learn Advanced Linear Algebra and Matrix Theory. But the books recommended seem again to treat space and matrix in separated chapters (and the level is less advanced than assumed here).

wpzdm
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    There is a set of notes called Linear algebra done wrong which strikes a very nice balance between these two approaches right the way through. It may be more basic than what you're looking for - it deals mostly with $\mathbb{R}^n$ and $\mathbb{C}^n$, and I don't think quotient spaces feature at all. – Joppy Sep 13 '21 at 08:52
  • @Joppy Thank you. It is indeed less advanced than I'd like to have. But maybe I can give it a try by directly jumping to Sec 5.5 or so. Could you point to an example (better in that latter part) where the two approaches are exploited to enlighten the topic so that I can get a taste? – wpzdm Sep 13 '21 at 10:23
  • Why the downvote? – wpzdm Sep 14 '21 at 02:48

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