Using SageMath

Now that we’ve learned the basics of row reduction to RREF, let’s explore how to perform row reduction on a computer. There are lots of viable tools for this purpose. Among the many possibilities that we could use, we will mostly stick with SageMath.

SageMath (or just Sage) is free, open-source mathematical software built on top of Python. You might think of it as Python plus a huge number of open source libraries built in. For those familiar with Python, the basic language and idioms should seem familiar. In addition, though, there are a great many commands for performing mathematical tasks that are pre-imported into the namespsace.

First steps with Sage

There are several ways to use Sage. By far the easiest, though, is via the Sage Cell Server. If you navigate to that webpage, you’ll find a code input box where you can just type code and press “Evaluate”.

Even better, it’s easy to embed those code boxes into any webpage. So, for example, you can press the evaluate button below to add 2 and 3:

The box is editable so that you can change it to add 3 and 4 or to solve differential equations or whatever. Here, for example, is how to solve a non-linear system of equations:

Now, that we see a couple of real solutions, we can plot those curves together with the solutions:

A few observations

• Sage is a symbolic system; that is, it can give you solutions like \[i\sqrt{2}/2 \text{ rather than } 0.7071i.\]
• I defined x,y = var('x,y') in both cells because definitions don’t generally persist between sage cells.
• If sage can solve non-linear systems symbolically, then I suppose you could use it to solve linear systems symbolically as well.

In spite of Sage’s symbolic capabilities, there are often good reasons to work numerically and to work with matrices.

Manipulating matrices with Sage

With Sage, we can define matrices and perform quite a few computations quickly and easily. Suppose, for example, we’d like to solve a system of equations:

\[ \begin{aligned} x+2y+3z &= 1 \\ 3x+2y-z &= -1 \\ 5x+2y-5z &= 1 \end{aligned} \]

We can use Sage’s built in matrix command to express the system as an augmented matrix and then use that matrix’s rref method to quickly reduce it to reduced row echelon form:

From there, of course, it’s easy to see that there’s an infinite family of solutions to to write down that family in parameterized form. Right?

Give it a try!

I’ve set up a 10 point problem on our forum so that everyone can try their hand at this. Give it a try!