High school student here...
Recently I was told I could go to the book store and pick out any math books I want. (2 or 3)
Does anyone have some good suggestions? I'm comfortable with anything involving calculus and I am currently studying linear algebra. It doesn't really even need to be a textbook just something that will help me learn and build mathematical maturity.
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V. Prasolov's Problems and Theorems in Linear Algebra will not only teach you everything you need to know about linear algebra, but also give you germs of ideas that will prove very useful later if you study more advanced mathematics where linear algebra occurs in a way or another.
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Two suggestions:
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If you wanna study linear algebra, then I'd like to suggest - I) Linear Algebra (Friedberg, Insel, Spence) II) Introduction to Linear Algebra (Gilbert Strang) III) Linear Algebra (Kwak and Hong) IV) Linear Algebra (Kenneth Hoffman, Ray Kunze)
For abstract algebra, go for - I) A First Course in Abstract Algebra (J.B. Fraleigh) II) Abstract Algebra (Dummit and Foote) III) Contemporary Abstract Algebra (Joseph A. Gallian)
For Analysis, study - I) Mathematical Analysis (Tom M. Apostol) II) Real and Complex Analysis (Walter Rudin) III) Real Analysis (Royden , Fitzpatrick) IV) Introduction to Real Analysis (Bartle, Sherbert)
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Vector Analysis by Murray Spiegel is one good book with lots of exercises. Also I googled "Cambridge University Maths Reading List" and found this. I am sure others exist if you look.
I read Fraleigh's "First Course in Abstract Algebra" when I was at school, and found it accessible and good.
Körner's "Calculus for the ambitious" is a quirky approach - but geared to building understanding in the bridge between "calculus" at school and "analysis" at university.
And Feynman's Lectures on Physics are worth reading, too.
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I have a favorite author. Peter J. Cameron. I love the way he tells the math-story for you. He has some lecture notes that you may find on his homepage for both linear algebra and abstract algebra.
Beside that I recently faced a cute book for Linear Algebra which I highly recommend. The approach of the book is based on Linear Maps (and not matrices) and try to not use Determinant concept directly. The name of the book is "linear algebra done right".
Also, for real analysis I recommend to read Real Mathematical Analysis by Charles Pugh. The book is compact and lovely with really nice exercises.
Have you seen What is Mathematics? by Courant and Robbins? It is by far the best introduction to mathematics that you can get your hands on, in my opinion. Here's a short form of the table of contents for your reference: