feat: 1522 l1 notes

ma1522/info.yml

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+title: 'Linear Algebra for Computing'
+short: 'MA1522'

ma1522/lec_01.tex

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+\subsection{Linear Algebra}
+\begin{itemize}
+  \item \textbf{Linear} The study of items/planes and objects which are flat
+  \item \textbf{Algebra} Objects are not as simple as numbers
+\end{itemize}
+
+\subsection{Linear Systems \& Their Solutions}
+
+Points on a straight line are all the points $(x, y)$ on the $xy$ plane satisfying the linear eqn: $ax + by = c$, where $a, b > 0$
+
+\subsubsection{Linear Equation}
+Linear eqn in $n$ variables (unknowns) is an eqn in the form
+$$ a_1x_1 + a_2x_2 + ... + a_nx_n = b$$
+where $a_1, a_2, ..., a_n, b$ are constants. 
+
+\begin{note}
+In a linear system, we don't assume that $a_1, a_2, ..., a_n$ are not all 0
+\begin{itemize}
+  \item If $a_1 = ... = a_n = 0$ but $b \neq 0$, it is \textbf{inconsistent}
+
+    E.g. $0x_1 + 0x_2 = 1$
+
+  \item If $a_1 = ... = a_n = b = 0$, it is a \textbf{zero equation}
+
+    E.g. $0x_1 + 0x_2 = 0$
+  \item Linear equation which is not a zero equation is a \textbf{nonzero equation}
+
+    E.g. $2x_1 - 3x_2 = 4$
+  \item The following are not linear equations
+    \begin{itemize}
+      \item $xy = 2$
+      \item $\sin\theta + \cos\phi = 0.2$
+      \item $x_1^2 + x_2^2 + ... + x_n^2 = 1$
+      \item $x = e^y$
+    \end{itemize}
+\end{itemize}
+\end{note}
+
+In the $xyz$ space, linear equation $ax + by + cz = d$ where $a, b, c > 0$ represents a plane
+
+\subsubsection{Solutions to a Linear Equation}
+Let $a_1x_1 + a_2x_2 + ... + a_nx_n = b$ be a linear eqn in n variables \\
+For real numbers $s_1+ s_2+ ... + s_n$, if $a_1s_1 + a_2s_2 + ... + a_ns_n = b$, then $x_1 = s_1, x_2 = s_2, x_n = s_n$ is a solution to the linear equation \\
+The set of all solutions is the \textbf{solution set}\\
+Expression that gives the entire solution set is the \textbf{general solution}
+
+\textbf{Zero Equation} is satified by any values of $x_1, x_2,... x_n$
+
+General solution is given by $(x_1, x_2, ..., x_n) = (t_1, t_2, ..., t_n)$
+
+
+\subsubsection{Examples: Linear equation $4x-2y = 1$}
+\begin{itemize}
+  \item x can take any arbitary value, say t
+  \item $x = t \Rightarrow y = 2t - \frac{1}{2}$
+  \item General Solution: 
+    $
+    \begin{cases}
+      x = t & \text{t is a parameter}\\
+      y = 2t - \frac{1}{2}
+    \end{cases}
+    $
+  \item y can take any arbitary value, say s
+  \item $y = s \Rightarrow x = \frac{1}{2}s + \frac{1}{4}$
+  \item General Solution: 
+    $
+    \begin{cases}
+      y = s & \text{s is a parameter}\\
+      x = \frac{1}{2}s + \frac{1}{4}
+    \end{cases}
+    $
+\end{itemize}
+
+\subsubsection{Example: Linear equation $x_1 - 4x_2 + 7x_3 = 5$}
+
+\begin{itemize}
+  \item $x_2$ and $x_3$ can be chosen arbitarily, $s$ and $t$
+  \item $x_1 = 5 + 4s -7t$
+  \item General Solution: 
+    $
+    \begin{cases}
+      x_1 = 5 + 4s -7t \\
+      x_2 = s & s, t \text{ are arbitrary parameters}\\
+      x_3 = t \\
+    \end{cases}
+    $
+\end{itemize}
+
+
+\subsection{Linear System}
+Linear System of m linear equations in n variables is
+\begin{equation}
+\begin{cases}
+  a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n = b_1 \\
+  a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n = b_2 \\
+  \vdots \\
+  a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n = b_m \\
+\end{cases}
+\end{equation}
+where $a_{ij}, b$ are real constants and $a_{ij}$ is the coeff of $x_j$ in the $i$th equation
+
+\begin{note} Linear Systems
+  \begin{itemize}
+    \item If $a_{ij}$ and $b_i$ are zero, linear system is called a \textbf{zero system}
+    \item If $a_{ij}$ and $b_i$ is nonzero, linear system is called a \textbf{nonzero system}
+    \item If $x_1 = s_1, x_2 = s_2, ..., x_n = s_n$ is a solution to \textbf{every equation} in the system, then its a solution to the system
+    \item If every equation has a solution, there might not be a solution to the system
+    \item \textbf{Consistent} if it has at least 1 solution
+    \item \textbf{Inconsistent} if it has no solutions
+  \end{itemize}
+\end{note}
+
+
+\subsubsection{Example}
+\begin{equation}
+\begin{cases}
+a_1x + b1_y = c_1 \\
+a_2x + b2_y = c_2 \\
+\end{cases}
+\end{equation}
+
+where $a_1, b_1, a_2, b_2$ not all zero
+
+In $xy$ plane, each equation represents a straight line, $L_1, L_2$
+
+\begin{itemize}
+  \item If $L_1, L_2$ are parallel, there is no solution
+  \item If $L_1, L_2$ are not parallel, there is 1 solution
+  \item If $L_1, L_2$ coinside(same line), there are infinitely many solution
+\end{itemize}
+
+\begin{equation}
+\begin{cases}
+a_1x + b1_y + c_1z = d_1 \\
+a_2x + b2_y + c_2z = d_2 \\
+\end{cases}
+\end{equation}
+
+where $a_1, b_1, c_1, a_2, b_2, c_2$ not all zero
+
+In $xyz$ space, each equation represents a plane, $P_1, P_2$
+
+\begin{itemize}
+  \item If $P_1, P_2$ are parallel, there is no solution
+  \item If $P_1, P_2$ are not parallel, there is $\infty$ solutions (on the straight line intersection)
+  \item If $P_1, P_2$ coinside(same plane), there are infinitely many solutions
+  \item Same Plane $\Leftrightarrow a_1 : a_2 = b_1 : b_2 = c_1 : c_2 = d_1: d_2$
+  \item Parallel Plane $\Leftrightarrow a_1 : a_2 = b_1 : b_2 = c_1 : c_2$
+  \item Intersect Plane $\Leftrightarrow a_1 : a_2, b_1 : b_2, c_1 : c_2$ are not the same
+\end{itemize}
+
+\subsection{Augmented Matrix}
+$
+  \begin{amatrix}{3}
+    a_{11} & a_{12} & a_{1n} & b_1 \\
+    a_{21} & a_{12} & a_{2n} & b_2 \\
+    a_{m1} & a_{m2} & a_{mn} & b_m \\
+  \end{amatrix}
+$
+
+\subsection{Elementary Row Operations}
+To solve a linear system we perform operations: 
+\begin{itemize}
+  \item Multiply equation by nonzero constant
+  \item Interchange 2 equations
+  \item add a constant multiple of an equation to another
+\end{itemize}
+
+
+\begin{itemize}
+  \item Multiply row by nonzero constant
+  \item Interchange 2 rows
+  \item add a constant multiple of a row to another row
+\end{itemize}

ma1522/master.tex

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+\documentclass[a4paper]{article}
+
+\input{./preamble.tex}
+
+
+
+% ------------------------------------------------------------------------------
+
+\begin{document}
+
+% ------------------------------------------------------------------------------
+% Cover Page and ToC
+% ------------------------------------------------------------------------------
+
+\title{\normalsize \textsc{}
+		\\ [2.0cm]
+		\HRule{1.5pt} \\
+		\LARGE \textbf{\uppercase{MA1522}
+		\HRule{2.0pt} \\ [0.6cm] \LARGE{Linear Algebra in Computing} \vspace*{10\baselineskip}}
+		}
+\date{}
+\author{\textbf{Yadunand Prem}}
+
+\maketitle
+\newpage
+
+\tableofcontents
+\newpage
+
+% ------------------------------------------------------------------------------
+
+\section{Lecture 1}
+\hr
+\input{lec_01.tex}
+
+\newpage
+
+\section{Lecture 2}
+\hr
+\input{lec_02.tex}
+\section{Lecture 3}
+\hr
+\input{lec_03.tex}
+\section{Lecture 4}
+\hr
+\input{lec_04.tex}
+\section{Lecture 5}
+\hr
+\input{lec_05.tex}
+\section{Lecture 6}
+\hr
+\input{lec_06.tex}
+
+\section{Reference}
+
+\begin{theorem}
+    This is a theorem.
+\end{theorem}
+
+\begin{proposition}
+    This is a proposition.
+\end{proposition}
+
+\begin{principle}
+    This is a principle.
+\end{principle}
+
+\begin{note}
+This is a note
+\end{note}
+
+% Maybe I need to add one more part: Examples.
+% Set style and colour later.
+
+
+
+\newpage
+
+% ------------------------------------------------------------------------------
+
+\end{document}

ma1522/preamble.tex

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+\usepackage{amsmath, amsthm, amssymb, amsfonts}
+\usepackage{thmtools}
+\usepackage{graphicx}
+\usepackage{setspace}
+\usepackage{geometry}
+\usepackage{float}
+\usepackage{hyperref}
+\usepackage[utf8]{inputenc}
+\usepackage[english]{babel}
+\usepackage{framed}
+\usepackage[dvipsnames]{xcolor}
+\usepackage{tcolorbox}
+
+\colorlet{LightGray}{White!90!Periwinkle}
+\colorlet{LightOrange}{Orange!15}
+\colorlet{LightRed}{Red!15}
+\colorlet{LightGreen}{Green!15}
+
+\newcommand{\HRule}[1]{\rule{\linewidth}{#1}}
+
+\declaretheoremstyle[name=Theorem,]{thmsty}
+\declaretheorem[style=thmsty,numberwithin=section]{theorem}
+\tcolorboxenvironment{theorem}{colback=LightGray}
+
+\declaretheoremstyle[name=Proposition,]{prosty}
+\declaretheorem[style=prosty,numberlike=theorem]{proposition}
+\tcolorboxenvironment{proposition}{colback=LightOrange}
+
+\declaretheoremstyle[name=Principle,]{prcpsty}
+\declaretheorem[style=prcpsty,numberlike=theorem]{principle}
+\tcolorboxenvironment{principle}{colback=LightGreen}
+
+\declaretheoremstyle[name=Note,]{notesty}
+\declaretheorem[style=notesty,numbered=no]{note}
+\tcolorboxenvironment{note}{colback=LightRed}
+
+\setstretch{1.2}
+\geometry{
+    textheight=9in,
+    textwidth=5.5in,
+    top=1in,
+    headheight=12pt,
+    headsep=25pt,
+    footskip=30pt
+}
+
+\newcommand\hr{
+    \noindent\rule[0.5ex]{\linewidth}{0.5pt}
+}
+
+\newenvironment{amatrix}[1]{%
+  \left(\begin{array}{@{}*{#1}{c}|c@{}}
+}{%
+  \end{array}\right)
+}