this year I finished calculus BC, which is the equivalent of calc 1 and 2. I feel as though I've completely mastered the material and I'm ready to start moving on before I enter university next year. The next math classes I'd be taking would be differential equations, calc 3 (multivariable calculus) and linear algebra and I'm looking to get a head start on them over the summer since math is what I really want to do. I don't want to buy any textbooks for them so does anyone know good online resources for these subjects? also I feel focusing on all this calculus business may be boring at times over the summer so I'm open to any other mathematical subjects that are interesting and I can delve into at any pace I want. anyone have ideas? these are some interesting lectures i'm looking at right now but I really want to have a strong foundation in these topics. http://www.academicearth.org/courses/linear-algebra http://www.academicearth.org/courses/differential-equations http://www.academicearth.org/courses/multivariable-calculus-1
I don't know about the other two, but I've watched some of Gilbert Strang's linear algebra lectures and they're quite good. My first exposure to linear algebra was a summer class (can it already be 20 years ago?) that used Strang's "Linear Algebra and Its Applications" as the text, which I enjoyed at the time: http://www.amazon.com/Linear-Algebra-Applications-Gilbert-Strang/dp/0030105676 I see that the online course includes three quizzes and the final exam, but I don't see any homework problems. (But check out the final exam: "closed book, ten wonderful problems.") You will probably want to find a good source of problems if you want to gain any mastery over the material. I don't know any good online sources offhand, but if you have access to a decent library you can always borrow Strang's book.
This sounds like a good idea and I think I'll actually work through this book. Are there solutions? Are there books similar to spivak for the other topics I listed that build up a strong foundation and focus on proofs? I'm really looking to prepare myself for undergraduate studies in math.