Refined Lecture Notes

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Your Name

Published

February 5, 2025

Introduction

This document contains the refined lecture notes. The goal is to enhance the mathematical and technical content while maintaining precise notation and academic rigor. Special attention will be given to identifying and filling in any gaps or unclear details in the original transcription.

Mathematical Concepts

Set Theory

A set is a well-defined collection of distinct objects, considered as an object in its own right.

Examples of sets include the set of natural numbers $\mathbb{N} = \{1, 2, 3, ...\}$, the set of integers $\mathbb{Z} = \{..., -2, -1, 0, 1, 2, ...\}$, and the set of real numbers $\mathbb{R}$.

Functions

A function $f$ from a set $A$ to a set $B$, denoted as $f: A \to B$, is a relation that associates each element $x \in A$ with exactly one element $f(x) \in B$.

Let $f: A \to B$ and $g: B \to C$ be two functions. The composition of $g$ with $f$, denoted as $g \circ f$, is a function from $A$ to $C$ such that $(g \circ f)(x) = g(f(x))$ for all $x \in A$.

Linear Algebra

A vector space is a set $V$ together with two operations, vector addition and scalar multiplication, that satisfy certain axioms.

Let $V$ be a vector space over a field $F$. Then for all vectors $u, v, w \in V$ and scalars $a, b \in F$, the following properties hold:

$u + v = v + u$ (Commutativity of addition)
$(u + v) + w = u + (v + w)$ (Associativity of addition)
There exists a zero vector $0 \in V$ such that $u + 0 = u$ (Additive identity)
For every $u \in V$, there exists an additive inverse $-u \in V$ such that $u + (-u) = 0$ (Additive inverse)
$a(u + v) = au + av$ (Distributivity of scalar multiplication over vector addition)
$(a + b)u = au + bu$ (Distributivity of scalar multiplication over field addition)
$a(bu) = (ab)u$ (Associativity of scalar multiplication)
$1u = u$ (Scalar multiplication identity)

Calculus

The derivative of a function $f(x)$ at a point $x$ is defined as the limit: \[f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\] provided the limit exists.

The Fundamental Theorem of Calculus establishes the relationship between differentiation and integration. It has two parts:

If $F(x)$ is an antiderivative of $f(x)$, then $\int_a^b f(x) dx = F(b) - F(a)$.
If $g(x) = \int_a^x f(t) dt$, then $g'(x) = f(x)$.

Algorithms

$r \gets a \mod b$ $a \gets b$ $b \gets r$ return $a$

Remarks

Throughout this document, we will use standard mathematical notation. For example, $\mathbb{N}$ will denote the set of natural numbers, $\mathbb{Z}$ the set of integers, $\mathbb{R}$ the set of real numbers, etc.

--- title: "Refined Lecture Notes" author: "Your Name" date: "2025-02-05" format: html: toc: true # Table of Contents toc-depth: 2 code-tools: true theme: cosmo # Or "journal" for Distill-like minimalism --- # Introduction ::: tcolorbox This document contains the refined lecture notes. The goal is to enhance the mathematical and technical content while maintaining precise notation and academic rigor. Special attention will be given to identifying and filling in any gaps or unclear details in the original transcription. ::: # Mathematical Concepts ## Set Theory ::: tcolorbox A set is a well-defined collection of distinct objects, considered as an object in its own right. ::: ::: tcolorbox Examples of sets include the set of natural numbers $\mathbb{N} = \{1, 2, 3, ...\}$, the set of integers $\mathbb{Z} = \{..., -2, -1, 0, 1, 2, ...\}$, and the set of real numbers $\mathbb{R}$. ::: ## Functions ::: tcolorbox A function $f$ from a set $A$ to a set $B$, denoted as $f: A \to B$, is a relation that associates each element $x \in A$ with exactly one element $f(x) \in B$. ::: ::: tcolorbox Let $f: A \to B$ and $g: B \to C$ be two functions. The composition of $g$ with $f$, denoted as $g \circ f$, is a function from $A$ to $C$ such that $(g \circ f)(x) = g(f(x))$ for all $x \in A$. ::: ## Linear Algebra ::: tcolorbox A vector space is a set $V$ together with two operations, vector addition and scalar multiplication, that satisfy certain axioms. ::: ::: tcolorbox Let $V$ be a vector space over a field $F$. Then for all vectors $u, v, w \in V$ and scalars $a, b \in F$, the following properties hold: 1. $u + v = v + u$ (Commutativity of addition) 2. $(u + v) + w = u + (v + w)$ (Associativity of addition) 3. There exists a zero vector $0 \in V$ such that $u + 0 = u$ (Additive identity) 4. For every $u \in V$, there exists an additive inverse $-u \in V$ such that $u + (-u) = 0$ (Additive inverse) 5. $a(u + v) = au + av$ (Distributivity of scalar multiplication over vector addition) 6. $(a + b)u = au + bu$ (Distributivity of scalar multiplication over field addition) 7. $a(bu) = (ab)u$ (Associativity of scalar multiplication) 8. $1u = u$ (Scalar multiplication identity) ::: ## Calculus ::: tcolorbox The derivative of a function $f(x)$ at a point $x$ is defined as the limit: $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$ provided the limit exists. ::: ::: tcolorbox The Fundamental Theorem of Calculus establishes the relationship between differentiation and integration. It has two parts: 1. If $F(x)$ is an antiderivative of $f(x)$, then $\int_a^b f(x) dx = F(b) - F(a)$. 2. If $g(x) = \int_a^x f(t) dt$, then $g'(x) = f(x)$. ::: # Algorithms ::::: tcolorbox :::: algorithm ::: algorithmic $r \gets a \mod b$ $a \gets b$ $b \gets r$ **return** $a$ ::: :::: ::::: # Remarks ::: tcolorbox Throughout this document, we will use standard mathematical notation. For example, $\mathbb{N}$ will denote the set of natural numbers, $\mathbb{Z}$ the set of integers, $\mathbb{R}$ the set of real numbers, etc. :::