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So I know that a similar question has probably been asked about alternatives or compliments to this book, but I think my situation is different enough to warrant slightly different advice. So I've just completed a course that used this text and I've read it pretty thoroughly for the most part, but I didn't learn everything as well as I wanted (maybe that's normal), probably because some days I was slightly less motivated than others and so my knowledge is better in some areas than others. I want to close the gaps obviously, but I'm not sure I want to just re-read the text, not for a few weeks anyway.

Could anybody suggest another book that treats linear algebra in a similar theoretical fashion (less emphasis on matrices), or should I just wait a week or two after the semester is over to re-charge and just re-read sections from this text?

Also, I've had two semesters of real analysis, and am planning to self study functional analysis over the summer, will this afford me a chance to plug a few of the holes? As in, do a lot of the more fundamental results in linear algebra also pop up in a functional analysis text? That way I could possibly kill two birds with one stone?

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    $\begingroup$ Lang's "Linear Algebra" is a nice book, written at a similar level as Axler's, with a somewhat different viewpoint and coverage of many topics which are not in Axler. Hoffman and Kunze is another option at this level; many people like it but I don't care for it much. You could also check the (free) "Linear Algebra Done Wrong": math.brown.edu/~treil/papers/LADW/LADW.html $\endgroup$ – Bungo May 12 '15 at 5:39
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My suggestions for more theoretical linear algebra texts are:

  • Hoffman and Kunze
  • Shilov (Dover, so more "classical" but still useful)
  • Lang

Since you've had some form of real analysis, definitely start reading about functional analysis. I recommend Kreyszig or "Intro to Hilbert Space" by Young, or if you're feeling more brave, Reed & Simon, Lax, or Rudin.

Also, I noticed that my understanding of linear algebra went way up by studying a bit of abstract algebra, so maybe check out a book on groups too. Matrix groups are an added bonus, because they tackle linear algebra and analysis (maybe look into this set of lecture notes).

Also also, never underestimate the power of numerical linear algebra as a way to motivate and learn the theory. Even if you're not in to numerical methods, you'll never learn quite so much as when you have to implement it for real. Trefethen and Bau is a standard text there.

There are some fundamental results that come up in both finite dimensional LA and functional analysis. The mathematical physics texts like Reed and Simon or Courant and Hilbert usually do a good job of emphasizing this. Some of the key ones are:

Less emphasized in finite dimensional linear algebra but very important is functional calculus.

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