It's the zero part that bothered me. The difference between zero and a very very very small quantity is not a big deal until it turns a finite number into something that is not a finite number, and then it's an infinitely big deal. So my brain got distracted wondering "is it really zero?" until I reminded myself that the important thing is that I knew what you meant :-)
There are two schools of thought in teaching linear algebra. One focuses on matrices, and the other takes the perspective of linear maps and vector spaces. The course I took in college was all about matrices, and I didn't understand the point at all. I hated it. When I reviewed my linear algebra for grad school, I got a book that took the abstract approach, and it felt a lot simpler. The linear algebra perspective on Fourier analysis is that the functions e^2πisx form a basis for a vector space of functions, just like (1, 0, 0), (0, 1, 0), and (0, 0, 1) form a basis for R^3. Any function in that vector space can be represented as a linear combination of the basis elements. That representation is the Fourier series of the function. There are a lot of technical details to figure out, such as which functions are in the space, exactly how to calculate the coefficients of the linear combination, and how to figure out if a given Fourier series converges, but intuitively you can say:
"The Fourier transform is simply a method of expressing a function (which is a point in
some infinite dimensional vector space of functions) in terms of the sum of its projections
onto a set of basis functions.[1]"
There's a similar description on Wikipedia with more detail [2].
The neat thing is that even though Fourier transforms is a complicated subject, even though I barely scraped by learning the basics fifteen years ago, and even though I couldn't do any real calculations today to save my life, this way of looking at it is so simple that I can't forget it. When I look at the equations I am quickly oriented: the series is a linear combination of functions, the functions are an orthogonal basis of a vector space, and the coefficients of the linear combination are obtained by projecting the function onto the elements of the basis. It's a good place to start if I ever need to learn something about Fourier transforms again someday. It's also a good complement to the concrete spatiotemporal intuition that the article provides.
There are two schools of thought in teaching linear algebra. One focuses on matrices, and the other takes the perspective of linear maps and vector spaces. The course I took in college was all about matrices, and I didn't understand the point at all. I hated it. When I reviewed my linear algebra for grad school, I got a book that took the abstract approach, and it felt a lot simpler. The linear algebra perspective on Fourier analysis is that the functions e^2πisx form a basis for a vector space of functions, just like (1, 0, 0), (0, 1, 0), and (0, 0, 1) form a basis for R^3. Any function in that vector space can be represented as a linear combination of the basis elements. That representation is the Fourier series of the function. There are a lot of technical details to figure out, such as which functions are in the space, exactly how to calculate the coefficients of the linear combination, and how to figure out if a given Fourier series converges, but intuitively you can say:
"The Fourier transform is simply a method of expressing a function (which is a point in some infinite dimensional vector space of functions) in terms of the sum of its projections onto a set of basis functions.[1]"
There's a similar description on Wikipedia with more detail [2].
The neat thing is that even though Fourier transforms is a complicated subject, even though I barely scraped by learning the basics fifteen years ago, and even though I couldn't do any real calculations today to save my life, this way of looking at it is so simple that I can't forget it. When I look at the equations I am quickly oriented: the series is a linear combination of functions, the functions are an orthogonal basis of a vector space, and the coefficients of the linear combination are obtained by projecting the function onto the elements of the basis. It's a good place to start if I ever need to learn something about Fourier transforms again someday. It's also a good complement to the concrete spatiotemporal intuition that the article provides.
[1] http://undergraduate.csse.uwa.edu.au/units/CITS4240/Lectures... [2] http://en.wikipedia.org/wiki/Hilbert_space#Fourier_analysis