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I read in a thread about Artin's Algebra, and that it covers abstract and linear algebra. With the linear algebra in Apostol's books, I think I should be able to read Artin. I've heard good things about Rudin, but i'm unsure if I should buy Real and Complex Analysis if it is too complicated, especially if I will need knowledge of complex variables for electromagnetism.

Any help is appreciated, i'm a very motivated student for this type of stuff, and i'm willing to work through the most rigorous books in order to understand the mathematical principles behind physics.