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.
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.

A Gram Schmidt Illustration using matlab
A simple QR example using matlab
A least squares matlab example

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. To run this demo, open the SVDexample.mlx using the matlab open tab on the editor and then use the toggle (Run all). mlx files can be fragile, so if things are not working, type SVDexample on the command prompt in matlab. The executable portion of the code will run. Make sure that the code and the jpg image are in the same directory and that matlab choice of directory is the same.

I am posting an Extra Credit Worksheet. You will also need The leontief.mlx (to run, type leontief in ,matlab, make sure that your matlab session reads the directory where the script has been stored. And The markov.mlx (similar instructions apply to run this code) . If you would like to get at most a 20% grade correction on your midterm, download the extra credit worksheet and the mlx files which load into matlab and hand in the completed worksheet no later than the last monday of the term. For the computational portion, you are not compelled to use matlab (but you'll have to put together codes from scratch.

Mathematica is very easy to learn and will be extremely useful in classes, and in your future engineering, or scientific career, or professional career.

Mathematica has a great many tutorials and help pages at its web site. If you've never used it, the very first thing to do is to view This beginner 2-part tutorial. After that you can go to the learning center and the knowledge base. For fun you should check out mathematica alpha.

Note: mathematica CDF files do not require you to have mathematica. You do need to get the CDF player, which is available for free from WOLFRAM , the company the produces Mathematica

A Least Squares Demo.

An SVD Image Compression Example.

A 2D SVD Example that emphasizes basic transformations.

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.