# Understanding the Fourier Transform

tags

Mathematics

The projection of vector **a** onto vector **b** is defined:
$$\operatorname{proj}_{\mathbf{v}} \mathbf{a} = \frac{\langle \mathbf{a}, \mathbf{b} \rangle}{|\mathbf{a}|^{2}} \mathbf{a}$$

Thus, if ${\mathbf{e_1}, \cdots, \mathbf{e_n}}$ is an *orthogonal* basis of a vector space, then

$$\mathbf{v}=\sum_{i=1}^n \operatorname{proj}_{\mathbf{e_i}}(\mathbf{v})$$

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:

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

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