1 Fourier Analysis

Learn it tonight holy shit ur cooked

vector spaces, subspaces, inner products, dot products that move into the complex realm

Vector Spaces (Linear Spaces)

the notion of a vector is an abstract notion, cartesian vectors are vectors, for sure, but we are very soon going to generalize this so that a "vector" can be a signal. its is now "an element within a space that satisfies certain properties"

Algebraic Properties

when we move to distance, norm, normal of a vector, we'll follow Tom Apostol

Axioms

So, these are the axioms of a vector space.

Closure Axioms

axioms 1 and 2

1. Closure under vector addition
   • our vector space will have a name, \(\mathbb{V}\)
     • (in math) for every (A) x, y element of V, theres a unique z element of V (called the sum of x and y) denoted by z = x + y
     • Ex: V = [Z -> R]
       • if x and y are in the set V
       • z = x + y <-> z[n] = x[n] + y[n] (A)n element of Z
2. Closure under scalar multiplication
   • (A)alpha element of R or C theres a unique z element of V s.t. z = alpha * x*

Vector addition

a3 - a6

Scalar Multiplication

a7-a10