User: Rude Clerk

R is Open Source

On: Open Source Math Software For Education?

Since you want both Windows and Linux, as well as support for physics and statistics in addition to mathematics, then I recommend R - http://www.r.project.org

Easy to install and test out, and great package tools are onboard.

Recent versions are outperforming Matlab in number crunching benchmarks.

For a symbolic package, I personally prefer Maple to Mathematica.