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.
