---
lecture: 4
date: 2025-01-16
---
# The Inverse Image
The Inverse Image uses a [[Inverse Function]] $f^{-1} (z)$ of $Z \subset Y$ written $f: x \to y$ is $f^{-1}(z) \equiv \{ x \in X \mid f(x) \in Z \}$.
# Complex Numbers
In [[Complex Numbers in Sets]], $\mathbb{C} = \mathbb{R} \times \mathbb{R}$ with usual addition of vectors
## Multiplying Vectors 
Multiply vectors by adding their angles multiplying their lengths.

$$z = a + i \times b = (a, b)$$
($b$ here can be seen as $(a, 0) + (0, b) = (a+0, 0+b)$)
$$b = (b, 0) \implies i \times b = (0, b)$$

$i \times z$ rotates $z$ $90\degree$ counterclockwise.
## Proposition
$\mathbb{C}$ is complete ([[Cauchy Sequence|cauchy]]) and [[Algebraically Complete#For Complex Numbers]].
# Metric Spaces
[[Metric Space#Definition]]
## Example
Discrete metric on X; $d(x, y) = \begin{cases}0, & \text{if}\ x=y\\ 1, & \text{if}\ x \neq y\end{cases}$
# Vector Spaces
- [[Normed Vector Space]]s
- [[Complex Vector Space]]s
## Example
$$V = R^n = R \times \dots \times R = \{  (x_{1}, \, \dots, x_{n}) | x_{i} \in \mathbb{R} \}$$
(where the length of $R \times \, \dots \times R$ has $n$ $\mathbb{R}$s.)
$$V = \mathbb{R}^2 : (x , \, y) + (z, w) \equiv (x + z, \, y + w)$$
$$a \times (x, \, y) \equiv (ax, \, ay)$$
## [[Linear Basis]] Example
$$u = 3v_{1} + 5v_{2} + iv_{3}$$
$$3 v_{1} \neq 2v_{2}$$
$$c_{1}v_{1} + c_{2}v_{2} = 0 \implies c_{1} = 0 = c_{2}$$
$$\implies 0 \times v_{1} + 0 \times v_{2} = 0$$

Continuing the [[#Vector Spaces#Example]] but for Linear bases
$$v_{1} = (1, \, 0, \, 0, \, 0, \, \dots)$$
$$v_{2} = (0, \, 1, \, 0, \, 0, \, \dots)$$
$$v_{3} = (0, \, 0, \, 1, \, 0, \, \dots)$$
and the way of writing this would be:
$$(x_{1}, \, \dots, \, x_{n}) = \sum^{n}_{i=1} x_{i} \times v_{i} = 0$$
$$\implies x_{i} = 0$$
## Proposition
Any [[Vector Space]] $V$ has a [[Linear Basis]], and every basis has the same cardinality referred to as the $dim(V)$ (dimension of $V$) of $V$.
### Proof
[[ACIT4330/Lectures/Lecture 3#Axiom of Choice|Lecture 3#Axiom of Choice]]