Matlab is a linear algebra based program that is more numerically-oriented. It is ideally suited for learning linear algebra, and was developed by a terrific linear algebra scientist, Cleve Moler. It is FREE TO OSU Students!
To start: This is an old but really simple 1st tutorial on matlab. Should take you 40 mins to go through these notes.
First matlab session. . After the primer, try the set of instructions on this page (this is a homework for a totally different class, ignore references to the other class).

Moler's Book. Uses matlab.

An SVD Demo . (I found this on line. I have to find the person who wrote it so that I can give him/her credit). You will need the "Lena" image. There is an amusing story behind Lena Sodderberg's connection to image processing (search it).

Another SVD Demo . You will need the "Mona Lisa" image.

There are free symbolic/computational programs, such as sage, R, macsyma and python.

Of these I would strongly recommend mastering Python.

Python is actually a meta-language with all sorts of capabilities, extremely useful (scripting, symbolics, numerics, etc); much more to learn but you can adopt python for a great many things. All of the above programs have strengths and weaknesses, but no one will fall short of delivering a great deal of productivity, for the price.