I am a college sophomore with double majors in mathematics and microbiology, and I have been doing independent research in the mathematical/computational biology, which really led me to love the mathematics and its phenomenal applications. I will be taking the theoretical linear algebra and multi-variable calculus on upcoiming semester and proceed with introductory analysis (Rudin's PMA) on next Fall along with the theoretical differential equations. I used the calculus textbooks called "Calculus and Analytic Geometry" by George Simmons and "A First Course in Calculus" by Serge Lang to pass the single-variable calculus. The theoretical linear algebra course (one I will be taking on next semester), teaches the proof-based linear algebra and the proof-methodology; its required textbook is one written by Friedberg & Insel & Spence. The multi-variable calculus is heavily on computational and use my university's course notes...In this case, should I self-study the Spivak/Apostol/Courant before advancing into the analysis like Rudin's PMA? I own the Apostol's two volumes and also Lang's Basic Mathematics but I do not think I will be able to study them fully befoe the end of Summer. Is it okay to jump toward the math analysis without completing the Apostol or Spivak? My current research utilizes a lot of differential equations, linear algebra, and numerical analysis. Is it okay to study for numerical analysis without taking the analysis courses like real, complex, and functional analysis? If so, could you recommend a good introductory/elementary textbook on numerical analysis?
Also about the theoretical linear algebra textbooks, which among those books complement well with Friedberg? Hoffman & Kunze, Axler, Lang's LA, or Lang's Introduction to LA? If Hoffman & Kunze better for a starter than Friedberg? I always love to use at least two textbooks on given topic. Also are books called "How to Prove It" by Velleman and "How to Solve It" by Polya good for learning the proof methodology and techniques?
Thank you very much for your time, and I apologize for asking too many questions especially on the books. I think it is important in mathematics to choose the right books to study for. I look forward to your advice!