Where Can I Find Resources for Learning Linear and Abstract Algebra?

[derek.basler]

Well, first of all I'm extremely interested in math. I'm a junior in high school right now and I am taking the highest math i can for my grade, which is pre-calculus honors and I am also teaching myself single variable calculus. I have read some of Road to Reality by Roger Penrose, which teaches some pretty funky math, but i would really like to learn Linear and Abstract Algebra. Does anyone know of a site or a book that could really give me a good introduction to this kind of math? thank you in advance!

 

[slider142]

Linear Algebra is an excellent way to learn mathematical proof, as most of the proofs are relatively straightforward applications of the definitions (ie., compared to analysis or number theory). Some courses focus mostly on matrix methods, while others are more abstract. Both viewpoints are good to know. For the abstract viewpoint, "Linear Algebra Done Right" by Sheldon Axler is an easy introduction to the subject. As long as you have been introduced to vectors, any readable text on matrix algebra should be fine. To get a more general view of modern algebra, Michael Artin's "Algebra" is a natural follow-up to Axler's linear algebra text.
By the way, once you've completed about half of Sheldon's text, you should easily be able to tackle vector calculus. A nice text with lots of applied examples and proper theory is "Vector Calculus, Linear algebra, and Differential Forms" by Hubbard/Hubbard. After the first 3 chapters, you should be able to tackle Spivak's "Calculus on Manifolds", which will introduce you to the language of differential geometry. At some point during this, you may also want to check out texts on basic real analysis, complex analysis, and topology to round out the mathematics needed for a good understanding of the rest of that book. After Spivak and a good text on topology, texts on differential geometry and differential topology should be accessible.
Hopefully, you will also have a professor's help in your studies, as experience is invaluable.

 

[derek.basler]

wow that's a good amount of reading. Thank you! what i really hope to get out of this is the ability to write a good proof and understand why stuff works. Its odd because i used to hate proofs in geometry freshman year, but now i really want to learn to write some. I always wonder why stuff works in math, and I tend not to give up until i understand. So I will look up a few of those books, and if anyone else has any others thatd be cool. thank you again!