generated from smyalygames/quartz
47 lines
2.0 KiB
Markdown
47 lines
2.0 KiB
Markdown
---
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lecture: 4
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date: 2025-01-16
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---
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# The Inverse Image
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The Inverse Image uses a [[Inverse Function]] $f^{-1} (z)$ of $Z \subset Y$ written $f: x \to y$ is $f^{-1}(z) \equiv \{ x \in X | f(x) \in Z \}$.
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# Complex Numbers
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In [[Complex Numbers]], $\mathbb{C} = \mathbb{R} \times \mathbb{R}$ with usual addition of vectors
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## Multiplying Vectors
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Multiply vectors by adding their angles multiplying their lengths.
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$$z = a + i \times b = (a, b)$$
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($b$ here can be seen as $(a, 0) + (0, b) = (a+0, 0+b)$)
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$$b = (b, 0) \implies i \times b = (0, b)$$
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$i \times z$ rotates $z$ $90\degree$ counterclockwise.
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## Proposition
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$\mathbb{C}$ is complete ([[Cauchy Sequence|cauchy]]) and [[Algebraically Complete#For Complex Numbers]].
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# Metric Spaces
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[[Metric Space#Definition]]
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## Example
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Discrete metric on X; $d(x, y) = \begin{cases}0, & \text{if}\ x=y\\ 1, & \text{if}\ x \neq y\end{cases}$
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# Vector Spaces
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- [[Normed Vector Space]]s
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- [[Complex Vector Space]]s
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## Example
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$$V = R^n = R \times \dots \times R = \{ (x_{1}, \, \dots, x_{n}) | x_{i} \in \mathbb{R} \}$$
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(where the length of $R \times \, \dots \times R$ has $n$ $\mathbb{R}$s.)
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$$V = \mathbb{R}^2 : (x , \, y) + (z, w) \equiv (x + z, \, y + w)$$
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$$a \times (x, \, y) \equiv (ax, \, ay)$$
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## [[Linear Basis]] Example
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$$u = 3v_{1} + 5v_{2} + iv_{3}$$
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$$3 v_{1} \neq 2v_{2}$$
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$$c_{1}v_{1} + c_{2}v_{2} = 0 \implies c_{1} = 0 = c_{2}$$
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$$\implies 0 \times v_{1} + 0 \times v_{2} = 0$$
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Continuing the [[#Vector Spaces#Example]] but for Linear bases
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$$v_{1} = (1, \, 0, \, 0, \, 0, \, \dots)$$
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$$v_{2} = (0, \, 1, \, 0, \, 0, \, \dots)$$
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$$v_{3} = (0, \, 0, \, 1, \, 0, \, \dots)$$
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and the way of writing this would be:
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$$(x_{1}, \, \dots, \, x_{n}) = \Sigma^{n}_{i=1} x_{i} \times v_{i} = 0$$
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$$\implies x_{i} = 0$$
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## Proposition
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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 $V$.
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### Proof
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[[ACIT4330/Lectures/Lecture 3#Axiom of Choice|Lecture 3#Axiom of Choice]] |