If you're looking for an understanding of what the DFT computes, sure. But the OP was more interested in how it works than what it does, so all of the Matlab in the world won't do much good. If you try to actually write out the DFT matrix, you're overdoing it for this purpose---that it is possible to write the transformation from frequency-domain to time-domain as a multiplication by an invertible matrix is enough, since then the DFT is then just the solution of a system of linear equations.
You don't need a deep knowledge of linear algebra for the finite-length discrete-time case, just the basics that everyone studying maths/science/engineering gets taught in the first couple of months of university. The point that I was trying to make was that for finite-length discrete-time signals, all that you're really doing is solving a system of linear equations like every man and his dog can do after a bit of training---that your coefficients happen to be uniformly-sampled complex exponentials doesn't really matter until you start thinking about efficient implementation.
The distinction I suppose is between conclusions that one can work out by inspection or immediate application of theorem, rather than by taking a detour through intermediate steps. It's probably not quite so cut-and-dry, but that's how I tend to see things. Such turns of phrase are just about part of the jargon.
It almost functions as a punctuation mark to reassure the paranoid (which sometimes is practically everyone) that "no, this next bit really is what it looks like".
"Kill the Wabbit, Kill the Wabbit, Kill the Wabbit!" -- Looney Tunes, "What's Opera Doc?" (1957, Chuck Jones)