265 lines
6.6 KiB
TeX
265 lines
6.6 KiB
TeX
\documentclass[a4paper]{article}
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\usepackage{fancyhdr}
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\usepackage{extramarks}
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\usepackage{amsmath}
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\usepackage{amsthm}
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\usepackage{amsfonts}
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\usepackage{tikz}
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\usepackage{pgfplots}
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\usepackage[plain]{algorithm}
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\usepackage{algpseudocode}
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\usepackage{enumitem}
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\usepackage{framed}
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\usepackage{multicol}
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\graphicspath{ {./images/} }
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\usetikzlibrary{automata,positioning}
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%
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% Basic Document Settings
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%
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\topmargin=-0.45in
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\evensidemargin=0in
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\oddsidemargin=0in
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\textwidth=6.5in
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\textheight=9.0in
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\headsep=0.25in
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\linespread{1.1}
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\pagestyle{fancy}
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\lhead{\hmwkAuthorName}
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\chead{\hmwkClass\ : \hmwkTitle}
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\rhead{\firstxmark}
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\lfoot{\lastxmark}
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\cfoot{\thepage}
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\renewcommand\headrulewidth{0.4pt}
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\renewcommand\footrulewidth{0.4pt}
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\setlength\parindent{0pt}
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%
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% Create Problem Sections
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%
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\newcommand{\enterProblemHeader}[1]{
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\nobreak\extramarks{}{Question \arabic{#1} continued on next page\ldots}\nobreak{}
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\nobreak\extramarks{Question \arabic{#1} (continued)}{Question \arabic{#1} continued on next page\ldots}\nobreak{}
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}
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\newcommand{\exitProblemHeader}[1]{
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\nobreak\extramarks{Question \arabic{#1} (continued)}{Question \arabic{#1} continued on next page\ldots}\nobreak{}
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\stepcounter{#1}
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\nobreak\extramarks{Question \arabic{#1}}{}\nobreak{}
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}
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\setcounter{secnumdepth}{0}
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\newcounter{partCounter}
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\newcounter{homeworkProblemCounter}
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\setcounter{homeworkProblemCounter}{1}
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\nobreak\extramarks{Question \arabic{homeworkProblemCounter}}{}\nobreak{}
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\newenvironment{homeworkProblem}[1][-1]{
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\ifnum#1>0
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\setcounter{homeworkProblemCounter}{#1}
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\fi
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\section{Question \arabic{homeworkProblemCounter}}
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\setcounter{partCounter}{1}
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\enterProblemHeader{homeworkProblemCounter}
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}{
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\exitProblemHeader{homeworkProblemCounter}
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}
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\newcommand{\hmwkClass}{INF 1004 Mathematics 2}
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\newcommand{\hmwkTitle}{Tutorial\ Save\ My\ Grades}
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\newcommand{\hmwkAuthorName}{\textbf{Woon Jun Wei}}
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\newcommand{\hmwkStudentID}{\textbf{2200624}}
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\title{
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\vspace{2in}
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\textmd{\textbf{\hmwkClass \\ \hmwkTitle}}\\
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\vspace{3in}
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}
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\author{\hmwkAuthorName \\ \hmwkStudentID}
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\date{\today}
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\renewcommand{\part}[1]{\textbf{\large Part \arabic{partCounter}}\stepcounter{partCounter}\\}
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%
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% Various Helper Commands
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%
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% Useful for algorithms
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\newcommand{\alg}[1]{\textsc{\bfseries \footnotesize #1}}
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% For derivatives
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\newcommand{\deriv}[1]{\frac{\mathrm{d}}{\mathrm{d}x} (#1)}
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% For partial derivatives
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\newcommand{\pderiv}[2]{\frac{\partial}{\partial #1} (#2)}
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% Integral dx
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\newcommand{\dx}{\mathrm{d}x}
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% Alias for the Solution section header
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\newcommand{\solution}{\textbf{\large My Solution\\}}
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\newcommand{\answer}{\textbf{\large Sample Solutions\\}}
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% Probability commands: Expectation, Variance, Covariance, Bias
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\newcommand{\E}{\mathrm{E}}
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\newcommand{\Var}{\mathrm{Var}}
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\newcommand{\Cov}{\mathrm{Cov}}
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\newcommand{\Bias}{\mathrm{Bias}}
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\def\therefore{\boldsymbol{\text{ }
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\leavevmode
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\lower0.4ex\hbox{$\cdot$}
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\kern-.5em\raise0.7ex\hbox{$\cdot$}
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\kern-0.55em\lower0.4ex\hbox{$\cdot$}
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\thinspace\text{ }}}
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\begin{document}
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\maketitle
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\pagebreak
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\begin{homeworkProblem}
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Remove vector $u=(-1,3,-4,2)$ from vector $v=(-2,2,2,5,6)$\\
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\solution
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\end{homeworkProblem}
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\pagebreak
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\begin{homeworkProblem}
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Note the difference in the order between remove from and project onto.\\
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Project vector $u=(-1,3,-4,2)$ onto vector $v=(3,-3,-1,1)$\\
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\solution
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\end{homeworkProblem}
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\pagebreak
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\begin{homeworkProblem}
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$$
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\begin{align*}
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x+4y+2z &=5.5\\
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-5x-22y-5z &= -45.5\\
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2x+4z+14z &=-25
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\end{align*}
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$$
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\begin{itemize}
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\item Show as an itermediate step the augmented matrix when for the first time the zeroth coulmn became a one-hot vector after performing transformations
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\item Show as an intermediate step the augmented matrix when for the first time the augmented matrix is in row echelon form.
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\item Write the set of all solutions as a single vector or a combination of vectors, None if there is no solution
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\end{itemize}
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\solution
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\end{homeworkProblem}
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\pagebreak
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\begin{homeworkProblem}
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$$
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\begin{align*}
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x+3y-5z &=2.75\\
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3x+12y-13z &=-9.75\\
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-4x-6z+25z &=-46.25 \text{(might be -6y)}
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\end{align*}
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$$
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\begin{itemize}
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\item Show as an itermediate step the augmented matrix when for the first time the zeroth coulmn became a one-hot vector after performing transformations
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\item Show as an intermediate step the augmented matrix when for the first time the augmented matrix is in row echelon form.
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\item Write the set of all solutions as a single vector or a combination of vectors, None if there is no solution
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\end{itemize}
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\solution
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\end{homeworkProblem}
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\pagebreak
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\begin{homeworkProblem}
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Compute the inverse of
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$$
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\begin{align*}
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A_0 &= \begin{bmatrix}9&-2\\3&-4\end{bmatrix}\\
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A_1 &= \begin{bmatrix}10&3\\8&4\end{bmatrix}\\
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\end{align*}
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$$
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Use these inverses to Solve
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$$
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\begin{align*}
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A_0x &= \begin{bmatrix}1\\-2\end{bmatrix}\\
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A_1x &= \begin{bmatrix}-7\\4\end{bmatrix}\\
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\end{align*}
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$$
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\begin{itemize}
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\item Are they invertible?
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\item Which of them has full rank? WHich one of them has lower rank and which one?
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\end{itemize}
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\solution
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\end{homeworkProblem}
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\pagebreak
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\begin{homeworkProblem}
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Compute the determinant of
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$$
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\begin{align*}
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A &= \begin{bmatrix}3&-1&4\\5&2.5&3\\1&8&-6\end{bmatrix}\\
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A &= \begin{bmatrix}3&-2&0.5\\2.5&-3&1\\3&2&4\end{bmatrix}\\
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A &= \begin{bmatrix}2&-2&2\\8&3&-2\\10&-4.5&5\end{bmatrix}\\
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\end{align*}
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$$
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\solution
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\end{homeworkProblem}
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\pagebreak
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\begin{homeworkProblem}
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What is the determinant of this matrix? Write it as a polynomial in $c$.\\
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For what value $c$ the matrix is not invertible?
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$$
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A = \begin{bmatrix}6&-3&c\\5&2&2\\-2&-6&-2\end{bmatrix}
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$$
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\solution
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\end{homeworkProblem}
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\pagebreak
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\begin{homeworkProblem}
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Compute and apply the Householder matrix which makes transforms the first column of $A$ to a multipile of the first one-hot vector for
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$$
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A = \begin{bmatrix}8&1&2\\4&-1&3\\-8&4&2\end{bmatrix}
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$$
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and for (Subtracting is nicer)\\
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$$
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A = \begin{bmatrix}3&-4&3\\\sqrt{2}&6&4\\\sqrt{5}&3&2\end{bmatrix}
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$$
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\end{homeworkProblem}
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\pagebreak
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\end{document}
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