CONTEXT:
I am mathematics and software engineering student, currently in my second year of study. Due to personal circumstances I had to take a leave of absence from my studies for a period of 1.5 years, during which I did not have the opportunity to study any mathematics, and I have forgotten a lot of what I had already learned. The last mathematics courses I have done were introductory single variable calculus, computational linear algebra (with some theory thrown in there, equivalent to what you can find in the first chapter of Friedberg, Insel, Spence's linear algebra book), and discrete mathematics (geared towards computer science majors). In all three courses, I noticed that I liked the proof aspect a lot more than I did any of the applications. I should also mention that at the time, I did not know that I will enjoy the more abstract side of math so I did not attempt the equivalent proof based/theory-heavy courses.
In about 5 months, I will have to either take a rigorous multivariable calculus course (which will also cover a short introduction to real analysis) meant for pure math majors or an applications based course. If I pick the applications based one, then I will not be able to major in pure maths (I'll have to do applied instead), as my lack of familiarity with mathematical rigour and proofs will prevent me from doing subsequent courses that are proof-heavy (advanced linear algebra, abstract algebra, and real and complex analysis, advanced combinatorics and number theory and others).
Question:
I would like to use all the free time I have (which is about 4-5 hours a day, for about 5 months) to try to refresh my mathematics knowledge and go through a rigours single variable calculus and a proof based introduction to linear algebra textbooks. However, I am not sure what books would be worth investing my time into going through given the time frame I have. I own a copy of Spivak's calculus but from what I read online, it does not seem feasible to go through the entire book (doing most of the problems) in 5 months, or is it? I also had a look at Apostol's calculus (vol 1), and the problems seem easier, and it covers more ground than Spivak. Given my situation, which out of the two books would you recommend? Do you have an alternative advice you would like to offer, perhaps a better way to get more familiar with constructing proofs while revising calculus at the same time? Any and all advice is appreciated.
Currently, I am going through David Poole's linear algebra book (about half way through) and should be done with it in about a month, after which I'm planning to give Friedberg, Insel, Spence's linear algebra book a shot. If you also think that there's a better alternative, please let me know.
A small note on the "advice" tag, I understand that my question does seem to be along the lines of "what 's the best..." , but I hope that the context I provided allows you to formulate a more concrete response and if it does not, then please let me know, and I will remove the question.
Regarding the single-variable calculus part, how about giving a try to "Understanding Analysis by Stephen Abbot", from what i heard (and used a bit for dedekind cuts) it's really a nice intro to analysis, well it covers what you need, rigorous single-variable calc, proof based and a plus point it's not that big, i mean like Spivak or Apostol 1 (i did some chapters from mr Apostol). Imo though, i really like "Real Mathematical Analysis" by Pugh, you can also give it a go, the author's writing style is a fun one, i love it :).
For linear algebra, if you want to take a look, P.R. Halmos "Finite Dimensional vector spaces" or "Linear Algebra" by Hoffman & Kunze are good ones. I guess there's lang "linear algebra" (undergraduate one) too.