Linear algebra alone isn't enough, you also need complex geometry for Euler's formula - I can't just look at exp(0+xi) and go "hey, that's cos x+sin x i". And even then, you need to go a little way into linear algebra to encounter basis functions. Finally, you pretty much need someone to assert orthogonality to even consider it.

And if you aren't working with wavefunctions regularly, like physicists and engineers might be, that gets obtuse fast (I'm referring to the use of the FT in econometrics and image processing). Then it's not at all clear that you should even want a sinusoidal basis with integer frequencies.