Hi!
I'm looking for calculus book or any other resources of your recommendation to learn it. Most of the things that can be found easily and those that I learned at uni are crap. Usually they don't stick to one notation or there is a lot of hand waving instead of proofs. I'm looking for something formal and strict, yet readable and well structured. I know more or less what the main ideas are but I would like to learn it from scratch.
I know that we have a lot of very smart people here on Codeforces, so I believe that some of you might know something worth sharing/ recommendation, especially people who feel deep understanding of the topic. It would be also great if someone can propose some materials for learning (linear) algebra from scratch. So basically my request extends for both of those subjects.
Thanks in advance :)
You can watch 3Blue1Brown's "Essence of Calculus" and "Essence of Linear Algebra" series. Hope it helps ;-)
They are very nice, but they are still basically only visual companions. OP said they wanted something formal and strict, which 3b1b is not.
That's because Calculus (as it is taught in universities) really isn't a proof course. The main objective of these courses is that people can operate with and calculate limits, derivatives, integrals, diff. equations etc. Usually Calculus doesn't include actual proofs and definitions or if it does, they're just really half-assed.
If you really want to study the formal proofs in these topics, the thing you're looking for is an introductory course in Real Analysis. Unfortunately I don't have a good resource in English to recommend, but search for Real Analysis books, not Calculus ones.
How about "Linear Algebra Done Right" by Axler?
For Real Analysis, the standard text is Walter Rudin's "Principles of Mathematical Analysis," which rigorously develops the subject from the least upper bound property of the real numbers. Since it's widely used across many universities, there are plenty of solutions guides floating around online in the event that you get stuck on any of the exercises. A much gentler introduction can be found in Abbot's "Understanding Analysis," which covers many of the topics of the first book but limits the level of generality to the real line (rather than a general metric space).