That's because it's important that people understand the meaning of the Fourier Transform and what it actually calculates. It's not just some random basis you throw in an equation without understanding what you're doing.

You can't "throw a basis in an equation", that's not a known mathematical procedure.

Once you learn enough about mathematics, and here linear algebra, you will start to reduce more complicated concepts to more simple concepts to build the complex ideas up from simple ones. In this case, the notion of a change of basis is definitely much simpler than the notion of an integral transform, so it makes sense to express it that way.

"What it actually calculates" can be a tricky topic in mathematics. It calculates what we want it to calculate, that's how people came up with the definition. So the way to understand "what it calculates" is to understand "what do I want it to calculate". Somewhat like programming, really. If we want to calculate a function's representation in a convenient basis, then that's what we'll make Fourier transform do.

Thanks for the response! I've always wanted to have math explained to me by a 21-24 year old that spent too much time with Rudin and Axler in undergrad.

Were you planning on bringing anything of substance to this discussion?

Can't we just appreciate that both points of views are helpful? Instead of thinking that one is better, and saying that therefore, the other sucks.

Both of them explain different things. The "it's just a basis transformation" makes invertibility, additivity, etc. obvious. The spinning function things makes it clear why we want to do it and what it means to say, approximate by only the small frequencies. (The basis thing tells you nothing about why looking at just the low coefficients is helpful, sometimes. A basis doesn't care about ordering)

Except for orientability, I guess.

I agree- the Discrete Fourier Transform can be easily explained in terms of basic linear algebra. And that explanation is usually included in an introduction-level DSP book. Both Oppenheim and Proakis have a section devoted to this view- understanding the DFT as matrix multiplication starts you on the road to the FFT algorithm.

But most ECE undergrads will first see the Fourier transform in a continuous setting. Then it isn't so much an issue of "basic linear algebra"- students have almost certainly never talked about the space of square integrable functions, the inner product on this space, a basis for continuous functions, what it means for an infinite collection of objects to be orthogonal, etc... That is, if they have even taken any linear algebra yet!

Students learn to associate "Fourier Transform" with "complicated math" and then are discouraged when they learn about the DFT. At least, that has been my experience when TAing signal processing classes.

I have no idea about engineering, but I don't see how a physicist would not know this. Basis transformations in infinitely dimensional Hilbert spaces are an integral part of quantum mechanics. The Fourier-transform in particular is used for transformation between space |r> and momentum |p> representations, but it's also common to use energy representations |E_n> where the orthonormal basis is typically somewhat more exotic (e.g. given by a Gaussian multiplied with Hermite polynomials, as is the case for the QHO).

That's sad. It reminds me of "maths phobia", which is apparently a thing people can have. Also "maths anxiety".

I think part of it is also that physicists and engineers are the ones primarily teaching the subject. So it might not be that they can't handle the mathematics, but that they simply prefer to think entirely in intuition land where they can spin functions around without anyone protesting.