At IMTEK, I have been working with Mathematica and I should say that it is a pretty good tool for fast prototyping. The only disadvantage is that it is a commercial tool and it is necessary to purchase a license. As a result, when I finished my work at IMTEK, I have decided to look what is available for free. After some search, I have decided to switch to Python. Python does not have numerics as such, but to this end one can use NumPy and SciPy (www.scipy.org). I should say that commercial and free tools have their own advantages and disadvantages and everybody should take his own decision. Yet, I could live with NumPy and SciPy and actually I enjoy using them. After all for that price, one cannot complain.
I have used NumPy and SciPy in my library for postprocessing results of model reduction
and one can find there examples on how I have employed NumPy and SciPy to solve different linear algebra problems. Just search for numpy and scipy. Some other small examples are at