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I would like to know what are the best resources to use to teach and learn elementary subjects (calculus, linear algebra). I remember when learning calculus, I used Spivak's book, which had wonderful problems, but the explanations were not really memorable, the illustrations weren't that appealing and it didn't have enough applications. I would like to know the mathematician's opinion on Strang's books on calculus and linear algebra and the others. Are they not fit to be recommended to math majors? Are they rigorous on any level? What other books do you recommend that have what these don't?

Also, I think that learning through examples first and some visualisation is superior to putting generalities upfront with countless formulas Therefore if you know any book that's written in this vein, please post it.

P.S. Strang's books stress the importance of matrix notation. Is that good? Does it add any insight?

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    $\begingroup$ Strang's linear algebra books have great intuition and insights. They are certainly fit to be recommended to math majors, but I think math majors should also read books like Lax's linear algebra book which develop the theory in a more abstract setting and present more rigorous proofs. $\endgroup$ – littleO Aug 9 '14 at 19:27
  • $\begingroup$ Thanks,any comment on the other books? $\endgroup$ – Daniel Faust Montana Aug 10 '14 at 3:33
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    $\begingroup$ I'm not sure about Strang's book, but I would recommend checking out Apostol's Calculus, Volumes I and II. These volumes are FULL of applications, and have a good beginning outline of linear algebra. The rigor is on the level of Spivak. $\endgroup$ – Darrin Aug 10 '14 at 3:37
  • $\begingroup$ @Darrin I like Apostol a great deal,but it's rather dry and it's unorthodox organization may confuse the crap out of a beginner. It also costs a king's ransom now,sadly. $\endgroup$ – Mathemagician1234 Aug 10 '14 at 19:51
  • $\begingroup$ I've read apostol's books,it's way too dry. $\endgroup$ – Daniel Faust Montana Aug 12 '14 at 0:09
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I consider both of Strang's books must-reads for undergraduates. These are what applied mathematics books should look like. There's a perception that applied books are merely rote procedural texts, where proofs are omitted and they basically give toolbox approaches to the material. Bad applied books do this—good ones like Strang's don't omit proofs, but they merely downplay them. They are both rich with relevant applications—that is, applications that are currently being used in research and practice by applied mathematicians and other professionals—beautifully written and conceptually careful. The Calculus book has the added bonus of being available online for free at Strang's website. Don't be fooled into spending all your money on the second edition of the calculus book—it's no different from the first, it merely contains Strang's "Highlights of Calculus" in addition, which is good but doesn't really add anything to the book's quality.

A word of warning: There are several versions of Strang's linear algebra texts. The one you really want to read is Linear Algebra And Its Applications. The other book-Introduction To Linear Algebra-is a dramatically watered down version of this book and it's missing many of Strang's wonderful sidebars and digressions. So the "non-introduction" version is the one you want.

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  • $\begingroup$ Thanks,any comments on his applied math book? $\endgroup$ – Daniel Faust Montana Aug 10 '14 at 4:55
  • $\begingroup$ Thanks for clarifying that Linear Algebra and its Applications is the better one. That's the one I have always loved, but I've never read Introduction to Linear Algebra and I wasn't sure how it compared. I've noticed that Linear Algebra and its Applications is very expensive on amazon, but a pdf can be found online. Note that Strang has several other excellent books -- Introduction to Applied Math; Computational Science and Engineering; a book on wavelets I haven't read; and a Linear Algebra, Geodesy and GPS book which is great. $\endgroup$ – littleO Aug 10 '14 at 5:06
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    $\begingroup$ @littleO I should have clarified my comments. It's not about length per se, it was about the fact that INTRODUCTION doesn't cover anywhere near as many or as deep a selection of topics. It develops topics with more detail and at a lower level then LAWA. $\endgroup$ – Mathemagician1234 Aug 10 '14 at 5:31
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    $\begingroup$ @Daniel Faust Montana ABSOLUTELY as long as they aren't the only textbooks you ever use on the subject. A serious mathematics major will obviously need to supplement them with more theoretical treatments, such as Friedberg,Insel and Spence for linear algebra and Donald Estep,Arthur Mattuck's wonderful- and now affordable!-text or Lax/Terrell for calculus. For the record-while I love the depth and beauty of Spivak, I wonder if the austerity of the text is really appropriate for a beginner,even a very talented one. (continued) $\endgroup$ – Mathemagician1234 Aug 10 '14 at 19:47
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    $\begingroup$ @ Montana I like Zorich a great deal, but I think the beginner-even a very good one-is going to find it VERY hard. It also doesn't have anywhere near enough exercises for me. From what I've seen of it,Lax/Terrell looks outstanding-wish it was cheaper. $\endgroup$ – Mathemagician1234 Aug 11 '14 at 4:22

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