I was looking at the Mathematica website today and just decided to buy the software. I've spent too much time juggling annoying symbolic expressions by hand that actually learning to do things with a computer should be a worthwhile investment. Similarly for making conjectures about formulas etc. a computer would seem like a useful tool.
My question: Where should I begin? I have a Ph.D. in math and would consider myself an expert level programmer in C, C++, x86 assembler, OCaml and Perl. I understand the functional programming paradigm well, so I don't need any introductions. To begin with, I would like to be able to work efficiently with linear algebra, number theory and random sampling. Oh, and plotting would be nice too and how to embed formulas written in LaTeX into those plots.