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diff --git a/notes.tex b/notes.tex new file mode 100644 index 0000000..49cc068 --- /dev/null +++ b/notes.tex @@ -0,0 +1,195 @@ +\documentclass[a4paper]{article} + +% Imports +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{multicol} +\usepackage{ragged2e} +\usepackage{blindtext} +\usepackage[english]{babel} %this is the dictionary you will use +\usepackage{graphicx,mathdots,chemarr,fancyvrb,comment} %some more packages +\usepackage{tikz} %some more packages +%the packages from here on will help with creating a graph +%tikzpicture +\usepackage[many]{tcolorbox} +\usepackage{wrapfig} +\usepackage{scalerel} +\usepackage{pict2e} +\usepackage{tkz-euclide} +\usepackage{scalerel} +\usepackage{pict2e} +\usepackage{tkz-euclide} +\usepackage{listings} +\usepackage{color} +\usepackage{xifthen} +\usepackage{hyperref} +\graphicspath{ {/home/jliu/Documents/texassets/} } +\definecolor{dkgreen}{rgb}{0,0.6,0} +%\definecolor{gray}{rgb}{0.5,0.5,0.5} +\definecolor{red}{HTML}{ffb3b3} +\definecolor{redbar}{HTML}{ff0000} +\definecolor{mauve}{rgb}{0.58,0,0.82} +\definecolor{cyanbar}{HTML}{00bfff} +\definecolor{cyan}{HTML}{b3f0ff} +\definecolor{greenbar}{HTML}{00ff00} +\definecolor{green}{HTML}{8cd98c} +\definecolor{main}{HTML}{5989cf} % setting main color to be used +\definecolor{sub}{HTML}{cde4ff} % setting sub color to be used +\tcbset{ + sharp corners, + colback = white, + before skip = 0.2cm, % add extra space before the box + after skip = 0.5cm % add extra space after the box +} % setting global options for tcolorbox +\makeatletter + + +%My Custom Commands +\newcommand{\mnewline}{\newline\newline\newline} +\newcommand{\mline}{\rule{0.5cm}{0.5pt}} +\newcommand{\proj}[1]{\text{Proj}_{#1}} +\newcommand{\st}{\ni:} +\newcommand{\evaline}[2]{\Big|^{#1}_{#2}} +\newcommand{\nulli}[1]{\text{Null }{#1}} +\newcommand{\ran}[1]{\text{ran }{#1}} +\newcommand{\col}[1]{\text{Col }({#1})} +\newcommand{\re}[1]{\text{Re}({#1})} +\newcommand{\im}[1]{\text{Im}({#1})} +\newcommand{\spa}[1]{\text{span}\{{#1}\}} +%\newcommand{\neproof}[3]{$\text{Let } \epsilon > {#2} \text{ be given}$\\\text{Choose $N = {#1}$\text{ Suppose $n > N > {#3}$}}} +\newcommand{\neproof}[3]{ %The first one is without the 3rd argument and the second one is + \ifthenelse{\isempty{#3}}{$\text{Let } \epsilon > {#2} \text{ be given}$\\\text{Choose $N = {#1}$\text{ Suppose $n > N$}}} + {$\text{Let } \epsilon > {#2} \text{ be given}$\\\text{Choose $N = {#1}$\text{ Suppose $n > N > {#3}$}}} +} +\newcommand{\infobox}[2]{\begin{InfoBox} + \smash{\raisebox{-5pt}{\includegraphics[width=0.77cm,height=0.68cm]{information}}}{\bf #1}\newline\newline + {#2} +\end{InfoBox}} +\newcommand{\warningbox}[2]{\begin{WarningBox} + \smash{\raisebox{-6pt}{\includegraphics[width=0.70cm,height=0.70cm]{warning}}} + {\bf #1}\newline\newline + {#2} +\end{WarningBox}} +\newcommand{\theorybox}[2]{\begin{TheoryBox} + \smash{\raisebox{-6pt}{\includegraphics[width=0.70cm,height=0.70cm]{theorem}}} + {\bf #1}\newline\newline + {#2} +\end{TheoryBox}} +\newcommand{\notebox}[2]{\begin{NoteBox} + \smash{\raisebox{-6pt}{\includegraphics[width=0.55cm,height=0.70cm]{reminder}}} + {\bf #1}\newline\newline + {#2} +\end{NoteBox}} +\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{% + \hskip -\arraycolsep + \let\@ifnextchar\new@ifnextchar + \array{#1}} +\newtcolorbox{InfoBox}{ + colback = sub, + colframe = main, + boxrule = 0pt, + leftrule = 6pt % left rule weight +} +\newtcolorbox{WarningBox}{ + colback = red, + colframe = redbar, + boxrule = 0pt, + leftrule = 6pt % left rule weight +} +\newtcolorbox{TheoryBox}{ + colback = cyan, + colframe = cyanbar, + boxrule = 0pt, + leftrule = 6pt % left rule weight +} +\newtcolorbox{NoteBox}{ + colback = green, + colframe = greenbar, + boxrule = 0pt, + leftrule = 6pt % left rule weight +} +\newtcolorbox{TitleBox}{ + boxrule = 2pt, + rounded corners +} +\makeatother +\usepackage[letterpaper,left=6mm,includemp=true,marginparwidth=12mm,marginparsep=1mm,reversemarginpar,right=19mm, +includefoot=true,top=19mm,nohead,footskip=12mm,bottom=6mm]{geometry} +% Here are the custom commands I have created. They are increadibly retarded +% mnewline: creates 3 newlines +% mline: Creates a horizontal line +% proj: Creates a Proj with a suitable subscript - Takes an argument +% st: creates a ni and a : as the 'such that' +% evaline: creates a vertical line for evaluated definite integrals. First argument is upper limit, second is lower. - Takes two arguments +% nulli: creates a Null (with a whitespace) - Takes an argument +% col: creates a Col (with a whitespace) - Takes an argument +% ran: creates a ran (with a whitespace) - Takes an argument +% re: creates a Re() - Takes an argument +% im: creates a Im() - Takes an argument +% sp: creates a span{} - Takes an argument +% neproof: Creates a cookie cutter N-epsilon proof. First argument set's N's value and second argument sets epsilon greater than value and the third (optional) argument sets the n > N > value. IF YOU DO NOT WANT THE THIRD ARGUMENT YOU NEED AN EMPTY CURLY BRACKET +\begin{document} + \setlength{\parindent}{1cm} + \begin{center} + {\bf \Large Self-Study Linear Algebra} + \end{center} + \begin{TitleBox} + \begin{center} + {\bf bruh} + \end{center} + \end{TitleBox} + \theorybox{The center of it all}{Linear algebra, at it's core, is the study of linear maps on finite-dimensional vector spaces.} + \section{Foundational Knowledge} + \subsubsection{Lists} + \theorybox{Definition -- List -- Tuple}{A list is an ordered collection of non-negative {\bf finite} length of elements. Two lists are equal if and only if they contain the same elements in the same order and have the same length.} + \notebox{Lists are fucking finite}{Lists are finite. If you have an infinite number of elements, you do not have a list/tuple.} + \notebox{Notation}{Lists are surrounded by parenthesis, sets are surrounded by curly brackets.} + \section{Fields} + \theorybox{Definition -- Field}{A field is an algebric structure where the underlying set of objects contains at least two distinct elements, $0$ and $1$, along with two operations that satisfy seven total axioms. $0$ we usually refer to as zero or the additive identity, and $1$ we usually call one. The two operations, we usually refer to as addition and multiplication. Most people are familiar with them as $+$ and as $\times$. In this document, we shall refer the generalized addition as $\boxplus$ and generalized multiplication as $\boxdot$.\\ + The six axioms that must be satisfied for an algebraic structure to be considered a field are: + \begin{enumerate} + \item Addition \& Multiplication must be associative + \item Addition \& Multiplication must be commutative + \item There must exist a unique additive identity such that the sum of any element with the additive identity results in the original element + \item There must exist a unique additive inverse for each element such that the sum of said element with it's inverse is the additive identity + \item There must exist a multiplicative identity such the multiplication of any element with the identity is the original element + \item There must exist a unique multiplicative inverse for each element such that the multiplication of any element with the inverse gives the multiplicative identity + \item Multiplication must distribute over addition + \end{enumerate} + Do note that often the identity axioms are squished together thus forming 6 axioms rather than 7, but just remember that these must be satisfied.\\\\ + Also remember that it is important for both operations to be closed. If they are not closed, then we do not have a field. + } + \subsubsection{Proof that Additive Inverses are unique for complex numbers} + Let's assume towards a contradiction that additive inverses are not unique.\\ + Let $x \in \mathbb{C}$ and let $\alpha, \beta \in \mathbb{C}$ such that $\alpha \not= \beta$ and $x + \alpha = 0$ and $x + \beta = 0$.\\ + Let $x = a + bi$, $\alpha = c + di$ and $\beta = e + fi$ where $a, b, c, d, e, f \in \mathbb{R}$.\\ + We can observe that + \begin{equation} + x + \alpha = x + \beta + \end{equation} + Then + \begin{alignat*}{3} + &x + \alpha = x + \beta&\\ + \Rightarrow&a + bi + c + di + -a = a + bi + e + fi + -a&\text{by additive inverse of $\mathbb{R}$}\\ + \Rightarrow&bi + di + c - c = bi + fi + e + -c&\text{by additive inverse of $\mathbb{R}$}\\ + \Rightarrow&bi + di = bi + fi + e - c&\\ + \Rightarrow&di = fi + e - c&\text{(1)}\\ + \Rightarrow&di + c = fi + e + -c + c&\\ + \Rightarrow&di + c = fi + e&\text{As previously established, $-c$ is the additive inverse of $c$}\\ + \Rightarrow&\alpha = \beta&\\ + \end{alignat*} + And thus we have arrived at a contradiction, thus additive inverses are unique.\\\\ + Note that (1) can be justified by adding the additive inverse of $bi$ to both sides. + \subsubsection{Proof that Multiplicative Inverses are unique for complex numbers} + Let $x \in \mathbb{C}$ and let $\alpha, \beta \in \mathbb{C}$ such that $\alpha x = 1$ and $\beta x = 1$.\\ + Then + \begin{alignat*}{3} + &\alpha&\qquad\\ + =&1 \alpha&\qquad\text{by multiplicative identity}\\ + =&\beta x \alpha&\qquad\text{by def}\\ + =&\beta (x \alpha)&\qquad\text{since multiplication is associative}\\ + =&\beta 1&\qquad\text{by def}\\ + =&\beta&\qquad\\ + \end{alignat*} + Thus $\alpha = \beta$, and thus multiplicative inverses are unique. +\end{document} |