Mathematica for linear algebra course?

Question

I'm taking a linear algebra / matrix theory course and we are free to use any software we want, and will be "expected to use MATLAB or an equivalent" for homework. The professor and textbook (Applied Linear Algebra) prefer MATLAB, some students seem to like R, but I'm already familiar with Mathematica and have a student license.

Are there any possible limitations of Mathematica to watch out for, if used as a learning device (i.e. not for work or intensive computation) for a linear algebra course? e.g. Are there some operations which the other software packages can do which Mathematica cannot out-of-the-box?

Answer 1

Mathematica offers a pretty complete set of functionality for linear algebra, and it has improved in recent versions.

For example, since version 5, Mathematica has offered the generalised Schur decomposition (also known as the QZ decomposition). This certainly wasn't available in earlier versions. It handles sparse matrices and many other wrinkles. And if it isn't available, this example shows that it can take a lot less code than in many other languages, so you are less likely to get yourself tangled up in loop constructs and the like.

Rather than focus on limitations, I thought it worth mentioning that Mathematica offers something neither Matlab nor R can, that makes it ideal as a pedagogical tool: integrated symbolics. If you are learning linear algebra and you want to understand certain things, you can in many cases pass a symbolic matrix to the relevant function, and use the resulting output to gain a deeper understanding.

Here's a simple example:

aa = Array[a, {2, 2}]

(* {{a[1, 1], a[1, 2]}, {a[2, 1], a[2, 2]}}*)

Eigenvalues[aa]

(*{1/2 (a[1, 1] + a[2, 2] - Sqrt[
    a[1, 1]^2 + 4 a[1, 2] a[2, 1] - 2 a[1, 1] a[2, 2] + a[2, 2]^2]), 
 1/2 (a[1, 1] + a[2, 2] + Sqrt[
    a[1, 1]^2 + 4 a[1, 2] a[2, 1] - 2 a[1, 1] a[2, 2] + a[2, 2]^2])}*)

Of course, sometimes the symbolic output is pretty hard to trace through (consider SingularValueDecomposition[aa] for the aa defined above!), but in can be useful in some cases.

Some pages on the internet contain claims that Mathematica is slower than Matlab, or that Matlab is somehow "better" for numerical work (and implicitly, Mathematica is only good for a bit of symbolic solving or something). As this answer shows, this claim is no longer true. It was true until about version 5, but no longer.

Answer 2

Having taught linear algebra using both Mathematica and Matlab, I concur with what others have said that the Mathematica's features for linear algebra include all one might need for a course in undergraduate linear algebra. Since symbolic computation is also fully integrated into Mathematica, it might be better in some ways. For example, we can solve symbolic systems of small dimension fairly easily.

LinearSolve[{{a, b}, {c, d}}, {e, f}]

Note how the determinant appears in the denominator illustrating its importance in determining when a system is solvable. This can be done just as easily for a 3x3 or 4x4 system.

Also, there is a common mis-conception that Mathematica does not or cannot distinguish between row and column vectors, as illustrated by Nasser's comment. I think this is not correct. It's just that if you want to represent a row or column vector, you should use the full matrix representation of said vector. Consider, for example, the following two dot product computations (whose orders cannot be reversed):

{{a, b}, {c, d}}.{{x}, {y}} // MatrixForm

{{x, y}}.{{a, b}, {c, d}} // MatrixForm

In addition, though, Mathematica provides a consistent notion of dot product between dimensions based on tensor products. Here's the product of a rank 3 tensor with a rank 2 tensor.

Table[a[i, j, k], {i, 1, 2}, {j, 1, 2}, {k, 1, 2}].
  Table[b[i, j], {i, 1, 2}, {j, 1, 2}]

By contrast, Matlab is limited to 2D floating point matrices. When you type x=2 in Matlab, you've defined a 2D matrix; the equivalent definition in Mathematica would be x={{2}}.

Answer 3

In years past I've taught a standard-content sophomore-level (in U.S.) linear algebra course where students used Mathematica. I know, then, that if you're interested, or if it's a requirement, you can build everything up from the simplest functions for manipulating matrices or you can directly use powerful built-in Mathematica functions (or a combination of both).

E.g., you might begin by defining little utility functions to scale a row, to swap two rows, and to add one row to another; from that, you can build a function to do Gauss-Jordan row reduction. Or you can directly use RowReduce.

Again, you might define a function to test for invertibility and, when invertible, find the inverse of a matrix by adjoining an identity matrix, row reducing, etc. Or you can directly use Inverse.

Even for something more complicated like Gram-Schmidt orthonormalization, you can build up a function for this yourself, beginning by defining a function that projects a vector onto the span of a given set of vectors. Or use Orthogonalize.

You could even define your own determinant function (in one of several ways), then use that to find eigenvalues, etc.

If your course involves advanced operations such as LU or QR decompositions, again you can define functions for these yourself or else use the built-in ones.

Of course in general you would expect built-in functions to handle round-off issues more robustly than anything you would program yourself. But this would be the same with Matlab.

So yes, Mathematica would be an entirely appropriate tool in place of Matlab.