# Understanding the Fourier Transform

- tags
- Mathematics

The projection of vector \(\mathbf{a}\) onto vector \(\mathbf{b}\) is defined: \[\textit{proj}_{\mathbf{v}} \mathbf{a} = \frac{\langle \mathbf{a}, \mathbf{b} \rangle}{|\mathbf{a}|^{2}} \mathbf{a}\]

Thus, if \(\{e_i\}_{i=1}^n\) is an *orthogonal* basis of a vector space, then

\[\mathbf{v}=\sum_i\textit{proj}_{e_i}{v}\]

As it turns out, functions are vectors. This is because vectors are simply mathematical objects that can be added together and multiplied by scalars. That is to say, because you can take *linear combinations* of factors, they are vectors.

We also define the *inner product* of two periodic functions \(s_1(t), s_2(t)\) as:

\[\langle s_1(t),s_2(t)\rangle = \frac{1}{T}\int_T s_1(t)s_2^{*}(t) dt \]

\[ |s(t)|^2 = \frac{1}{T}\int_T |s(t)|^2 dt\]