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+\graphicspath{ {./images/} } + +\usetikzlibrary{automata,positioning} + +% +% Basic Document Settings +% + +\topmargin=-0.45in +\evensidemargin=0in +\oddsidemargin=0in +\textwidth=6.5in +\textheight=9.0in +\headsep=0.25in + +\linespread{1.1} + +\pagestyle{fancy} +\lhead{\hmwkAuthorName} +\chead{\hmwkClass\ : \hmwkTitle} +\rhead{\firstxmark} +\lfoot{\lastxmark} +\cfoot{\thepage} + +\renewcommand\headrulewidth{0.4pt} +\renewcommand\footrulewidth{0.4pt} + +\setlength\parindent{0pt} + +% +% Create Problem Sections +% + +\newcommand{\enterProblemHeader}[1]{ + \nobreak\extramarks{}{Question \arabic{#1} continued on next page\ldots}\nobreak{} + \nobreak\extramarks{Question \arabic{#1} (continued)}{Question \arabic{#1} continued on next page\ldots}\nobreak{} +} + +\newcommand{\exitProblemHeader}[1]{ + \nobreak\extramarks{Question \arabic{#1} (continued)}{Question \arabic{#1} continued on next page\ldots}\nobreak{} + \stepcounter{#1} + \nobreak\extramarks{Question \arabic{#1}}{}\nobreak{} +} + +\setcounter{secnumdepth}{0} +\newcounter{partCounter} +\newcounter{homeworkProblemCounter} +\setcounter{homeworkProblemCounter}{1} +\nobreak\extramarks{Question \arabic{homeworkProblemCounter}}{}\nobreak{} + +\newenvironment{homeworkProblem}[1][-1]{ + \ifnum#1>0 + \setcounter{homeworkProblemCounter}{#1} + \fi + \section{Question \arabic{homeworkProblemCounter}} + \setcounter{partCounter}{1} + \enterProblemHeader{homeworkProblemCounter} +}{ + \exitProblemHeader{homeworkProblemCounter} +} + +\newcommand{\hmwkClass}{INF 1004 Mathematics 2} +\newcommand{\hmwkTitle}{Tutorial\ Save\ My\ Grades} +\newcommand{\hmwkAuthorName}{\textbf{Woon Jun Wei}} +\newcommand{\hmwkStudentID}{\textbf{2200624}} + +\title{ + \vspace{2in} + \textmd{\textbf{\hmwkClass \\ \hmwkTitle}}\\ + \vspace{3in} +} + +\author{\hmwkAuthorName \\ \hmwkStudentID} +\date{\today} + +\renewcommand{\part}[1]{\textbf{\large Part \arabic{partCounter}}\stepcounter{partCounter}\\} + +% +% Various Helper Commands +% + +% Useful for algorithms +\newcommand{\alg}[1]{\textsc{\bfseries \footnotesize #1}} + +% For derivatives +\newcommand{\deriv}[1]{\frac{\mathrm{d}}{\mathrm{d}x} (#1)} + +% For partial derivatives +\newcommand{\pderiv}[2]{\frac{\partial}{\partial #1} (#2)} + +% Integral dx +\newcommand{\dx}{\mathrm{d}x} + +% Alias for the Solution section header +\newcommand{\solution}{\textbf{\large My Solution\\}} +\newcommand{\answer}{\textbf{\large Sample Solutions\\}} + +% Probability commands: Expectation, Variance, Covariance, Bias +\newcommand{\E}{\mathrm{E}} +\newcommand{\Var}{\mathrm{Var}} +\newcommand{\Cov}{\mathrm{Cov}} +\newcommand{\Bias}{\mathrm{Bias}} + +\def\therefore{\boldsymbol{\text{ } +\leavevmode +\lower0.4ex\hbox{$\cdot$} +\kern-.5em\raise0.7ex\hbox{$\cdot$} +\kern-0.55em\lower0.4ex\hbox{$\cdot$} +\thinspace\text{ }}} + +\begin{document} + +\maketitle + +\pagebreak + +\begin{homeworkProblem} + Remove vector $u=(-1,3,-4,2)$ from vector $v=(-2,2,2,5,6)$\\ + + \solution + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Note the difference in the order between remove from and project onto.\\ + Project vector $u=(-1,3,-4,2)$ onto vector $v=(3,-3,-1,1)$\\ + + \solution + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + + $$ + \begin{align*} + x+4y+2z &=5.5\\ + -5x-22y-5z &= -45.5\\ + 2x+4z+14z &=-25 + \end{align*} + $$ + + \begin{itemize} + \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 + \item Show as an intermediate step the augmented matrix when for the first time the augmented matrix is in row echelon form. + \item Write the set of all solutions as a single vector or a combination of vectors, None if there is no solution + \end{itemize} + + \solution + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + + $$ + \begin{align*} + x+3y-5z &=2.75\\ + 3x+12y-13z &=-9.75\\ + -4x-6z+25z &=-46.25 \text{(might be -6y)} + \end{align*} + $$ + + \begin{itemize} + \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 + \item Show as an intermediate step the augmented matrix when for the first time the augmented matrix is in row echelon form. + \item Write the set of all solutions as a single vector or a combination of vectors, None if there is no solution + \end{itemize} + + \solution + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Compute the inverse of + $$ + \begin{align*} + A_0 &= \begin{bmatrix}9&-2\\3&-4\end{bmatrix}\\ + A_1 &= \begin{bmatrix}10&3\\8&4\end{bmatrix}\\ + \end{align*} + $$ + Use these inverses to Solve + $$ + \begin{align*} + A_0x &= \begin{bmatrix}1\\-2\end{bmatrix}\\ + A_1x &= \begin{bmatrix}-7\\4\end{bmatrix}\\ + \end{align*} + $$ + \begin{itemize} + \item Are they invertible? + \item Which of them has full rank? WHich one of them has lower rank and which one? + \end{itemize} + + \solution + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + Compute the determinant of + $$ + \begin{align*} + A &= \begin{bmatrix}3&-1&4\\5&2.5&3\\1&8&-6\end{bmatrix}\\ + A &= \begin{bmatrix}3&-2&0.5\\2.5&-3&1\\3&2&4\end{bmatrix}\\ + A &= \begin{bmatrix}2&-2&2\\8&3&-2\\10&-4.5&5\end{bmatrix}\\ + \end{align*} + $$ + + \solution + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + What is the determinant of this matrix? Write it as a polynomial in $c$.\\ + For what value $c$ the matrix is not invertible? + + $$ + A = \begin{bmatrix}6&-3&c\\5&2&2\\-2&-6&-2\end{bmatrix} + $$ + + \solution + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Compute and apply the Householder matrix which makes transforms the first column of $A$ to a multipile of the first one-hot vector for + + $$ + A = \begin{bmatrix}8&1&2\\4&-1&3\\-8&4&2\end{bmatrix} + $$ + and for (Subtracting is nicer)\\ + + $$ + A = \begin{bmatrix}3&-4&3\\\sqrt{2}&6&4\\\sqrt{5}&3&2\end{bmatrix} + $$ + +\end{homeworkProblem} +\pagebreak + +\end{document} diff --git a/Revision/Revision_Tut_Solutions.pdf b/Revision/Revision_Tut_Solutions.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c43d802ada37e7b09c91c5d5bd56fad5536134f9 GIT binary patch literal 142352 zcma%?Q;=xOvaZ{TP;=y<&se;~lie{dO)jY_&=PaF{S*?@=|RlCSzXVx&03>F;Q6 zZZ-ZEcr|YB7`xb_i;uMGX*lt6TY!dDVax#&F*8i>8u?yo(D-`q)o2#34^5m%sX+- 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+\lfoot{\lastxmark} +\cfoot{\thepage} + +\renewcommand\headrulewidth{0.4pt} +\renewcommand\footrulewidth{0.4pt} + +\setlength\parindent{0pt} + +% +% Create Problem Sections +% + +\newcommand{\enterProblemHeader}[1]{ + \nobreak\extramarks{}{Question \arabic{#1} continued on next page\ldots}\nobreak{} + \nobreak\extramarks{Question \arabic{#1} (continued)}{Question \arabic{#1} continued on next page\ldots}\nobreak{} +} + +\newcommand{\exitProblemHeader}[1]{ + \nobreak\extramarks{Question \arabic{#1} (continued)}{Question \arabic{#1} continued on next page\ldots}\nobreak{} + \stepcounter{#1} + \nobreak\extramarks{Question \arabic{#1}}{}\nobreak{} +} + +\setcounter{secnumdepth}{0} +\newcounter{partCounter} +\newcounter{homeworkProblemCounter} +\setcounter{homeworkProblemCounter}{1} +\nobreak\extramarks{Question \arabic{homeworkProblemCounter}}{}\nobreak{} + +\newenvironment{homeworkProblem}[1][-1]{ + \ifnum#1>0 + \setcounter{homeworkProblemCounter}{#1} + \fi + \section{Question \arabic{homeworkProblemCounter}} + \setcounter{partCounter}{1} + \enterProblemHeader{homeworkProblemCounter} +}{ + \exitProblemHeader{homeworkProblemCounter} +} + +\newcommand{\hmwkClass}{INF 1004 Mathematics 2} +\newcommand{\hmwkTitle}{Revision\ Tutorial\ Solutions} +\newcommand{\hmwkAuthorName}{\textbf{Woon Jun Wei}} +\newcommand{\hmwkStudentID}{\textbf{2200624}} + +\title{ + \vspace{2in} + \textmd{\textbf{\hmwkClass \\ \hmwkTitle}}\\ + \vspace{3in} +} + +\author{\hmwkAuthorName \\ \hmwkStudentID} +\date{\today} + +\renewcommand{\part}[1]{\textbf{\large Part \arabic{partCounter}}\stepcounter{partCounter}\\} + +% +% Various Helper Commands +% + +% Useful for algorithms +\newcommand{\alg}[1]{\textsc{\bfseries \footnotesize #1}} + +% For derivatives +\newcommand{\deriv}[1]{\frac{\mathrm{d}}{\mathrm{d}x} (#1)} + +% For partial derivatives +\newcommand{\pderiv}[2]{\frac{\partial}{\partial #1} (#2)} + +% Integral dx +\newcommand{\dx}{\mathrm{d}x} + +% Alias for the Solution section header +\newcommand{\solution}{\textbf{\large My Solution\\}} +\newcommand{\answer}{\textbf{\large Sample Solutions\\}} + +% Probability commands: Expectation, Variance, Covariance, Bias +\newcommand{\E}{\mathrm{E}} +\newcommand{\Var}{\mathrm{Var}} +\newcommand{\Cov}{\mathrm{Cov}} +\newcommand{\Bias}{\mathrm{Bias}} + +\def\therefore{\boldsymbol{\text{ } +\leavevmode +\lower0.4ex\hbox{$\cdot$} +\kern-.5em\raise0.7ex\hbox{$\cdot$} +\kern-0.55em\lower0.4ex\hbox{$\cdot$} +\thinspace\text{ }}} + +\begin{document} + +\maketitle + +\pagebreak + +\begin{homeworkProblem} + Remove vector $u=(-1,3,-4,2)$ from vector $v=(-2,2,2.5,6)$\\ + + \answer + + $$v_{new} = v - \frac{v\cdot u}{u\cdot u}u$$ + + $$ + \begin{align*} + u\cdot u &= (-1,3,-4,2)\cdot (-1,3,-4,2) = 30\\ + v\cdot u &= (-2,2,2.5,6)\cdot (-1,3,-4,2) = 10\\ + \end{align*} + $$ + + $$ + \begin{align*} + v_{new} &= (-2,2,2.5,6) - \frac{10}{30}(-1,3,-4,2) \\ + &= (-2,2,2.5,6) +(\frac{1}{3},-1,\frac{4}{3}, -\frac{2}{3})\\ + &= (-\frac{5}{3},1,\frac{13}{6},\frac{16}{3}) + \end{align*} + $$ + + You can check: + $$ + v_{new}\cdot u = (-\frac{5}{3},1,\frac{13}{6},\frac{16}{3})\cdot (-1,3,-4,2) = 0 + $$ + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Note the difference in the order between remove from and project onto.\\ + Project vector $u=(-1,-3,-4,2)$ onto vector $v=(3,-3,-1,1)$\\ + + \answer + + $$u_{new} = \frac{u\cdot v}{v\cdot v}v$$ + + $$ + \begin{align*} + u_{new} &= \frac{(-1,-3,-4,2)\cdot (3,-3,-1,1)}{(3,-3,-1,1)\cdot (3,-3,-1,1)}(3,-3,-1,1)\\ + &= \frac{12}{20}(3,-3,-1,1)\\ + \end{align*} + $$ + + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + + $$ + \begin{align*} + x+4y+2z &=5.5\\ + -5x-22y-5z &= -45.5\\ + 2x+4y+14z &=-25 + \end{align*} + $$ + + \begin{itemize} + \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 + \item Show as an intermediate step the augmented matrix when for the first time the augmented matrix is in row echelon form. + \item Write the set of all solutions as a single vector or a combination of vectors, None if there is no solution + \end{itemize} + + \solution + + $$ + \begin{align*} + &\left[ + \begin{array}{ccc|c} + 1 & 4 & 2 & 5.5\\ + -5 & -22 & -5 & -45.5\\ + 2 & 4 & 14 & -25 + \end{array} \right] \\ + \rho_2+5\rho_1, \rho_3-2\rho_1 &\rightarrow \left[ + \begin{array}{ccc|c} + 1 & 4 & 2 & 5.5\\ + 0 & -2 & 5 & -18\\ + 0 & -4 & 10 & -36 + \end{array} \right] \\ + \rho_3-2\rho_2 &\rightarrow \left[ + \begin{array}{ccc|c} + 1 & 4 & 2 & 5.5\\ + 0 & -2 & 5 & -18\\ + 0 & 0 & 0 & 0 + \end{array} \right] \\ + -\frac{1}{2}\rho_2&\rightarrow \left[ + \begin{array}{ccc|c} + 1 & 4 & 2 & 5.5\\ + 0 & 1 & -2.5 & 9\\ + 0 & 0 & 0 & 0 + \end{array} \right] \\ + \rho_1+4\rho_2 &\rightarrow \left[ + \begin{array}{ccc|c} + 1 & 0 & 12 & -30.5\\ + 0 & 1 & -2.5 & 9\\ + 0 & 0 & 0 & 0 + \end{array} \right] \\ + \end{align*} + $$ + + $$ + \begin{align*} + x + 12z &= -30.5\\ + y - 2.5z &= 9\\\\ + \therefore x &= -30.5 - 12z\\ + \therefore y &= 9 + 2.5z\\ + \therefore z &= 0 + 1z\\\\ + \begin{bmatrix} x\\ y\\ z \end{bmatrix} &= \begin{bmatrix} + -30.5\\9\\0\end{bmatrix} + \begin{bmatrix} -12\\2.5\\1\end{bmatrix}z + \end{align*} + $$ + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + + $$ + \begin{align*} + x+3y-5z &=2.75\\ + 3x+12y-13z &=-9.75\\ + -4x-6y+25z &=-46.25 + \end{align*} + $$ + + \begin{itemize} + \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 + \item Show as an intermediate step the augmented matrix when for the first time the augmented matrix is in row echelon form. + \item Write the set of all solutions as a single vector or a combination of vectors, None if there is no solution + \end{itemize} + + \answer + + + $$ + \left[ \begin{array}{ccc|c} + 1 & 3 & -5 & 2.75\\ + 3 & 12 & -13 & -9.75\\ + -4 & -6 & 25 & -46.25 + \end{array} \right] \\ + $$ + + $$ + \begin{align*} + \rho_2-3\rho_1, \rho_3+4\rho_1 &\rightarrow \left[ \begin{array}{ccc|c} + 1 & 3 & -5 & 2.75\\ + 0 & 3 & 2 & -18\\ + 0 & 6 & 5 & -35.25 + \end{array} \right] \\ + \rho_3-2\rho_2 &\rightarrow \left[ \begin{array}{ccc|c} + 1 & 3 & -5 & 2.75\\ + 0 & 3 & 2 & -18\\ + 0 & 0 & 1 & 0.75 + \end{array} \right] \\ + \frac{1}{3}\rho_2-2\rho_1 &\rightarrow \left[ \begin{array}{ccc|c} + 1 & 3 & -5 & 2.75\\ + 0 & 1 & 0 & -6.5\\ + 0 & 0 & 1 & 0.75 + \end{array} \right] \\ + \rho_1-3\rho_2+5\rho_1 &\rightarrow \left[ \begin{array}{ccc|c} + 1 & 0 & 0 & 26\\ + 0 & 1 & 0 & -6.5\\ + 0 & 0 & 1 & 0.75 + \end{array} \right] \\ + \end{align*} + $$ + + $$ + \begin{align*} + \text{first time one hot:}& \left [ \begin{array}{ccc|c} + 1 & 3&-5 & 2.75\\ + 0 & 3 & 2 & -18\\ + 0 & 6 & 5 & -35.25 + \end{array} \right ] \\ + \text{first time row echelon form:}& \left [ \begin{array}{ccc|c} + 1&3&-5&2.75\\ + 0&1&\frac{2}{3}&6\\ + 0&0&1&0.75 + \end{array} \right ] \\ + \text{Reduced row echelon form:}& \left[ \begin{array}{ccc|c} + 1&0&0&26\\ + 0&1&0&-6.5\\ + 0&0&1&0.75 + \end{array} \right ] \\ + \end{align*} + $$ + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Compute the inverse of + $$ + \begin{align*} + A_0 &= \begin{bmatrix}9&-2\\3&-4\end{bmatrix}\\ + A_1 &= \begin{bmatrix}10&3\\8&4\end{bmatrix}\\ + \end{align*} + $$ + Use these inverses to Solve + $$ + \begin{align*} + A_0x &= \begin{bmatrix}1\\-2\end{bmatrix}\\ + A_1x &= \begin{bmatrix}-7\\4\end{bmatrix}\\ + \end{align*} + $$ + + \answer + + \begin{multicols}{2} + + $$ + \begin{align*} + A_0 &= \begin{bmatrix}9&-2\\3&-4\end{bmatrix}\\ + \det(A_0) &= 9\cdot(-4) - (-2)\cdot3 =-36+6 = -30\\ + A_0^{-1} &= \frac{1}{-30}\begin{bmatrix}-4&2\\-3&-9\end{bmatrix}\\ + &= \begin{bmatrix}\frac{2}{15}&-\frac{1}{15}\\\frac{1}{10}&-\frac{3}{10}\end{bmatrix}\\ + x &= A_0^{-1}\begin{bmatrix}1\\-2\end{bmatrix}\\ + &= \begin{bmatrix}\frac{2}{15}&-\frac{1}{15}\\\frac{1}{10}&-\frac{3}{10}\end{bmatrix}\begin{bmatrix}1\\-2\end{bmatrix}\\ + &= \begin{bmatrix}\frac{2}{15}+\frac{2}{15}\\\frac{1}{10}+\frac{6}{10}\end{bmatrix}\\ + &= \begin{bmatrix}\frac{4}{15}\\\frac{7}{10}\end{bmatrix}\\ + \end{align*} + $$ + + \columnbreak + + $$ + \begin{align*} + A_1 &= \begin{bmatrix}10&3\\8&4\end{bmatrix}\\ + \det(A_1) &= 10\cdot4 - 3\cdot8 = 40-24 = 16\\ + A_1^{-1} &= \frac{1}{16}\begin{bmatrix}4&-3\\-8&10\end{bmatrix}\\ + &= \begin{bmatrix}\frac{1}{4}&-\frac{3}{16}\\\frac{-1}{2}&\frac{5}{8}\end{bmatrix}\\ + x &= \begin{bmatrix}\frac{1}{4}&-\frac{3}{16}\\\frac{-1}{2}&\frac{5}{8}\end{bmatrix} \begin{bmatrix}-7\\4\end{bmatrix}\\ + &= \begin{bmatrix}-\frac{10}{4}\\6\end{bmatrix}\\ + \end{align*} + $$ + + \end{multicols} + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + Compute the determinant of + $$ + \begin{align*} + A &= \begin{bmatrix}3&-1&4\\5&2.5&3\\1&8&-6\end{bmatrix}\\ + A &= \begin{bmatrix}3&-2&0.5\\2.5&-3&1\\3&2&4\end{bmatrix}\\ + A &= \begin{bmatrix}2&-2&2\\8&3&-2\\10&-4.5&5\end{bmatrix}\\ + \end{align*} + $$ + + \begin{itemize} + \item Are they invertible? + \item Which of them has full rank? Which one of them has lower rank and which one? + \end{itemize} + + \answer + + \begin{multicols}{3} + + $$ + \begin{align*} + A &= \begin{bmatrix}3&-1&4\\5&2.5&3\\1&8&-6\end{bmatrix}\\ + \det{A} &= 0 (\text{not invertible})\\ + \text{rank} &= 2\\ + \end{align*} + $$ + + $$ + \begin{align*} + A &= \begin{bmatrix}3&-2&0.5\\2.5&-3&1\\3&2&4\end{bmatrix}\\ + \det{A} &= -21 (\text{invertible})\\ + \text{rank} &= 3\\ + \end{align*} + $$ + + $$ + \begin{align*} + A &= \begin{bmatrix}2&-2&2\\8&3&-2\\10&-4.5&5\end{bmatrix}\\ + \det{A} &= 0 (\text{not invertible})\\ + \text{rank} &= 2\\ + \end{align*} + $$ + + \end{multicols} + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + What is the determinant of this matrix? Write it as a polynomial in $c$.\\ + For what value $c$ the matrix is not invertible? + + $$ + A = \begin{bmatrix}6&-3&c\\5&2&2\\-2&-6&-2\end{bmatrix} + $$ + + \answer + + $$ + \begin{align*} + \det(A) &= 30-30c+4c\\ + &= 30-26c\\ + \det(A) &= 0\\ + \therefore c &= \frac{30}{26}\\ + \end{align*} + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + \textbf{Compute and apply} the Householder matrix which makes transforms the first column of $A$ to a multipile of the first one-hot vector for + + $$ + A = \begin{bmatrix}8&1&2\\4&-1&3\\-8&4&2\end{bmatrix} + $$ + and for (Subtracting is nicer)\\ + + $$ + A = \begin{bmatrix}3&-4&3\\\sqrt{2}&6&4\\\sqrt{5}&3&2\end{bmatrix} + $$ + + \answer + + \begin{multicols}{2} + + \part + + $$ + \begin{align*} + x &=\begin{bmatrix}8\\4\\-8\end{bmatrix}\\ + u &= \begin{bmatrix}8\\4\\-8\end{bmatrix} \pm \lVert \begin{bmatrix}8,4,-8\end{bmatrix}\rVert \begin{bmatrix}1\\0\\0\end{bmatrix}\\ + &= \begin{bmatrix}-4\\4\\-8\end{bmatrix} \text{ Subtract}\\ + H_u &= I - \frac{2}{\lVert [-4,4,-8]\rVert_2^2}\begin{bmatrix}8\\4\\-8\end{bmatrix}\begin{bmatrix}8&4&8\end{bmatrix}\\ + &= \begin{bmatrix}\frac{2}{3}&\frac{1}{3}&-\frac{2}{3}\\\frac{1}{3}&\frac{2}{3}&\frac{2}{3}\\-\frac{2}{3}&\frac{2}{3}&-\frac{1}{3}\end{bmatrix}\\ + HA &= \begin{bmatrix}12 &-2.44 & 1\\0 & 2.33 & 4\\0&-2.7&0\end{bmatrix}\\ + \end{align*} + $$ + + \columnbreak + + \part + + $$ + \begin{align*} + x &= \begin{bmatrix}3\\\sqrt{2}\\ \sqrt{5}\end{bmatrix}\\ + u &= \begin{bmatrix}3\\\sqrt{2}\\ \sqrt{5}\end{bmatrix} \pm \lVert \begin{bmatrix}3,\sqrt{2},\sqrt{5}\end{bmatrix}\rVert \begin{bmatrix}1\\0\\0\end{bmatrix}\\ + &= \begin{bmatrix}-1\\\sqrt{2}\\ \sqrt{5}\end{bmatrix} \text{ Subtract}\\ + H_u &= I - \frac{2}{\lVert [-1,\sqrt{2},\sqrt{5}]\rVert_2^2}\begin{bmatrix}-1\\\sqrt{2}\\\sqrt{5}\end{bmatrix}\begin{bmatrix}-1&\sqrt{2}&\sqrt{5}\end{bmatrix}\\ + &= \begin{bmatrix}\frac{3}{4}&\frac{\sqrt{2}}{4}&\frac{\sqrt{5}}{4}\\\frac{\sqrt{2}}{4}&\frac{1}{2}&-\frac{\sqrt{2}\sqrt{5}}{4}\\\frac{\sqrt{5}}{4}&-\frac{\sqrt{5}\sqrt{2}}{4}&-\frac{1}{4}\end{bmatrix}\\ + HA &= \begin{bmatrix}4&0.8&4.78\\0&-0.79&1.48\\0&-7.73&-1.99\end{bmatrix}\\ + \end{align*} + $$ + + \end{multicols} + +\end{homeworkProblem} +\pagebreak + +\end{document} diff --git a/Revision/additionaltasks_t2_exam.pdf b/Revision/additionaltasks_t2_exam.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9b74b2fb81f006a648e33cfbaf2a25dab359f55c GIT binary patch literal 107893 zcmbrmbyVEzvMq|cySr;Q?ry;~1PSiJ-QC^Y-6goYyCg^;K!OJiu6uhF1`PY-9j3J0mLu0RaS7c~e_+XA3eQ z7l51WUoQk!DNAc-Q%5pZDQiP#QwdXJI}=j`At3}OXGc>*8wB^Y6KyTKZ&GN!%XM3n zA?e?`FEwXz30 zIOcBcJj2~S_3-08Ul9(%tAu0pCj~VITLd(bM;Jl8gRfIe8g%Z>#=bYo1`aH29C|o> z9(x|_x$4aTt#(c_3U}yYrTzRFgqojA6--76)qGSUk$6c}L||FdeY0WdkT$D`*a@C; z*trm6=s*#l2osENBx}$9XOLJax$9lzlT=boV`3;II?(xocLfApPZbIf0M4gh)(V9gKS&VKna1V 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+\newcommand{\hmwkClass}{INF 1004 Mathematics 2} +\newcommand{\hmwkTitle}{Tutorial\ \#9} +\newcommand{\hmwkAuthorName}{\textbf{Woon Jun Wei}} +\newcommand{\hmwkStudentID}{\textbf{2200624}} + +\title{ + \vspace{2in} + \textmd{\textbf{\hmwkClass \\ \hmwkTitle}}\\ + \vspace{3in} +} + +\author{\hmwkAuthorName \\ \hmwkStudentID} +\date{\today} + +\renewcommand{\part}[1]{\textbf{\large Part \arabic{partCounter}}\stepcounter{partCounter}\\} + +% +% Various Helper Commands +% + +% Useful for algorithms +\newcommand{\alg}[1]{\textsc{\bfseries \footnotesize #1}} + +% For derivatives +\newcommand{\deriv}[1]{\frac{\mathrm{d}}{\mathrm{d}x} (#1)} + +% For partial derivatives +\newcommand{\pderiv}[2]{\frac{\partial}{\partial #1} (#2)} + +% Integral dx +\newcommand{\dx}{\mathrm{d}x} + +% Alias for the Solution section header +\newcommand{\solution}{\textbf{\large My Solution\\}} +\newcommand{\answer}{\textbf{\large Sample Solutions\\}} + +% Probability commands: Expectation, Variance, Covariance, Bias +\newcommand{\E}{\mathrm{E}} +\newcommand{\Var}{\mathrm{Var}} +\newcommand{\Cov}{\mathrm{Cov}} +\newcommand{\Bias}{\mathrm{Bias}} + +\begin{document} + +\maketitle + +\pagebreak + +\begin{homeworkProblem} + Compute $A^\top A$ for + $$ + \begin{equation*} + A = \begin{bmatrix} + 2&-3\\ + -6&-9 + \end{bmatrix} + \end{equation*} + + \begin{equation*} + A = \begin{bmatrix} + 0&1&-1\\ + 2&0&2\\ + 4&3&1 + \end{bmatrix} + \end{equation*} + $$ + + \solution + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Compute the inverse of + $$ + \begin{equation*} + A_0 = \begin{bmatrix} + -2&-3\\ + -6&-4 + \end{bmatrix} + \end{equation*} + $$ + $$ + \begin{equation*} + A_1 = \begin{bmatrix} + 3&1\\ + -2&2 + \end{bmatrix} + \end{equation*} + $$ + + Use this inverses to solve + + $$ + \begin{equation*} + A_0 x = \begin{bmatrix} + 3\\ + 1 + \end{bmatrix} + \end{equation*} + $$ + $$ + \begin{equation*} + A_0 x = \begin{bmatrix} + 2\\ + -5 + \end{bmatrix} + \end{equation*} + $$ + + Note: it is not common to solve $Ax=b$ using matrix inversion.\\ + + Reasons: + \begin{itemize} + \item $Ax=b$ can be solvable when $A$ is not invertible. + \item It is often slower / more costly + \end{itemize} + + \solution +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Compute the determinant of + + $$ + \begin{equation*} + A = \begin{bmatrix} + 1&2&-1\\ + 2&0&2\\ + 4&3&1 + \end{bmatrix} + \end{equation*} + \begin{equation*} + A = \begin{bmatrix} + 2&-4&3\\ + 1&1&1\\ + 3&3&1 + \end{bmatrix} + \end{equation*} + \begin{equation*} + A = \begin{bmatrix} + 1&0&1\\ + 2&3&2\\ + -1&3&2 + \end{bmatrix} + \end{equation*} + \begin{equation*} + A = \begin{bmatrix} + 1&2&3\\ + -5&-10&-15\\ + 6&12&18 + \end{bmatrix} + \end{equation*} + $$ + + \begin{itemize} + \item Are they Invertible? + \item Which of them has full rank? Which of them has lower rank and which one? + \end{itemize} + + \solution +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + For what value $a$ the matrix is not invertible? + + $$ + \begin{equation*} + A = \begin{bmatrix} + 1&2&1\\ + 2&a&4\\ + -3&1&2 + \end{bmatrix} + \end{equation*} + $$ + + \solution +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Compute and apply the Householder matrix which makes transforms the first column of + to a multple of the first one-hot vector $\begin{bmatrix}1\\0\\0\end{bmatrix}$ for + + $$ + \begin{equation*} + A = \begin{bmatrix} + 1&2&-1\\ + 2&0&2\\ + 2&3&1 + \end{bmatrix} + \end{equation*} + $$ + + and for (hint: here subtracting is nicer) + + $$ + \begin{equation*} + A = \begin{bmatrix} + 1&-4&3\\ + 3&1&1\\ + \sqrt{6}&3&1 + \end{bmatrix} + \end{equation*} + $$ + + \solution + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Verify that + + $$ + \begin{equation*} + A = \begin{bmatrix} + 1&0&0\\ + 0&\cos(\alpha)&-\sin(\alpha)\\ + 0&\sin(\alpha)&\cos(\alpha) + \end{bmatrix} + \end{equation*} + $$ + + Satisfies being an orthogonal matrix. + + \solution +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + + What is the cosine of the angle between + + $$ + \left( 6,-6,-4,\sqrt{12} \right), \left( 6,4,2,\sqrt{25} \right)? + $$ + + \solution +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Another $3\times3$ affine system + + \begin{itemize} + \item show the intermediate result when the first column is the one hot vector $\begin{bmatrix}1&0&0\end{bmatrix}$ for the first time + \item show that the intermediate result when the matrix has row echelon form for the first time + \item get the solution + \end{itemize} + + $$ + \begin{align*} + 2x-3y+2z &=-4\\ + 7x+4.5y-1z &=16\\ + 4x+3y+z &=2 + \end{align*} + $$ + + \solution +\end{homeworkProblem} + + +\end{document} diff --git a/Tutorial 10/INF1004_W10_Tutorial_2200624_Done.pdf b/Tutorial 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+\oddsidemargin=0in +\textwidth=6.5in +\textheight=9.0in +\headsep=0.25in + +\linespread{1.1} + +\pagestyle{fancy} +\lhead{\hmwkAuthorName} +\chead{\hmwkClass\ : \hmwkTitle} +\rhead{\firstxmark} +\lfoot{\lastxmark} +\cfoot{\thepage} + +\renewcommand\headrulewidth{0.4pt} +\renewcommand\footrulewidth{0.4pt} + +\setlength\parindent{0pt} + +% +% Create Problem Sections +% + +\newcommand{\enterProblemHeader}[1]{ + \nobreak\extramarks{}{Question \arabic{#1} continued on next page\ldots}\nobreak{} + \nobreak\extramarks{Question \arabic{#1} (continued)}{Question \arabic{#1} continued on next page\ldots}\nobreak{} +} + +\newcommand{\exitProblemHeader}[1]{ + \nobreak\extramarks{Question \arabic{#1} (continued)}{Question \arabic{#1} continued on next page\ldots}\nobreak{} + \stepcounter{#1} + \nobreak\extramarks{Question \arabic{#1}}{}\nobreak{} +} + +\setcounter{secnumdepth}{0} +\newcounter{partCounter} +\newcounter{homeworkProblemCounter} +\setcounter{homeworkProblemCounter}{1} +\nobreak\extramarks{Question \arabic{homeworkProblemCounter}}{}\nobreak{} + +\newenvironment{homeworkProblem}[1][-1]{ + \ifnum#1>0 + \setcounter{homeworkProblemCounter}{#1} + \fi + \section{Question \arabic{homeworkProblemCounter}} + \setcounter{partCounter}{1} + \enterProblemHeader{homeworkProblemCounter} +}{ + \exitProblemHeader{homeworkProblemCounter} +} + +\newcommand{\hmwkClass}{INF 1004 Mathematics 2} +\newcommand{\hmwkTitle}{Tutorial\ \#10} +\newcommand{\hmwkAuthorName}{\textbf{Woon Jun Wei}} +\newcommand{\hmwkStudentID}{\textbf{2200624}} + +\title{ + \vspace{2in} + \textmd{\textbf{\hmwkClass \\ \hmwkTitle}}\\ + \vspace{3in} +} + +\author{\hmwkAuthorName \\ \hmwkStudentID} +\date{\today} + +\renewcommand{\part}[1]{\textbf{\large Part \arabic{partCounter}}\stepcounter{partCounter}\\} + +% +% Various Helper Commands +% + +% Useful for algorithms +\newcommand{\alg}[1]{\textsc{\bfseries \footnotesize #1}} + +% For derivatives +\newcommand{\deriv}[1]{\frac{\mathrm{d}}{\mathrm{d}x} (#1)} + +% For partial derivatives +\newcommand{\pderiv}[2]{\frac{\partial}{\partial #1} (#2)} + +% Integral dx +\newcommand{\dx}{\mathrm{d}x} + +% Alias for the Solution section header +\newcommand{\solution}{\textbf{\large My Solution\\}} +\newcommand{\answer}{\textbf{\large Sample Solutions\\}} + +% Probability commands: Expectation, Variance, Covariance, Bias +\newcommand{\E}{\mathrm{E}} +\newcommand{\Var}{\mathrm{Var}} +\newcommand{\Cov}{\mathrm{Cov}} +\newcommand{\Bias}{\mathrm{Bias}} + +\begin{document} + +\maketitle + +\pagebreak + +\begin{homeworkProblem} + Compute $A^\top A$ for + $$ + \begin{equation*} + A = \begin{bmatrix} + 2&-3\\ + -6&-9 + \end{bmatrix} + \end{equation*} + + \begin{equation*} + A = \begin{bmatrix} + 0&1&-1\\ + 2&0&2\\ + 4&3&1 + \end{bmatrix} + \end{equation*} + $$ + + \solution + + \part + + $$ + \begin{align*} + A &= \begin{bmatrix} + 2&-3\\ + -6&-9 + \end{bmatrix}\\ + A^\top &= \begin{bmatrix} + 2&-6\\ + -3&-9 + \end{bmatrix}\\ + A^\top A &= \begin{bmatrix} + 2&-6\\ + -3&-9 + \end{bmatrix} + \begin{bmatrix} + 2&-3\\ + -6&-9 + \end{bmatrix}\\ + &= \begin{bmatrix} + 40&48\\ + 48&100 + \end{bmatrix} + \end{align*} + $$ + + \part + + $$ + \begin{align*} + A &= \begin{bmatrix} + 0&1&-1\\ + 2&0&2\\ + 4&3&1 + \end{bmatrix}\\ + A^\top &= \begin{bmatrix} + 0&2&4\\ + 1&0&3\\ + -1&2&1 + \end{bmatrix}\\ + A^\top A &= \begin{bmatrix} + 0&2&4\\ + 1&0&3\\ + -1&2&1 + \end{bmatrix} + \begin{bmatrix} + 0&1&-1\\ + 2&0&2\\ + 4&3&1 + \end{bmatrix}\\ + &= \begin{bmatrix} + 20&12&8\\ + 12&10&2\\ + 8&2&6 + \end{bmatrix} + \end{align*} + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Compute the inverse of + $$ + \begin{equation*} + A_0 = \begin{bmatrix} + -2&-3\\ + -6&-4 + \end{bmatrix} + \end{equation*} + $$ + $$ + \begin{equation*} + A_1 = \begin{bmatrix} + 3&1\\ + -2&2 + \end{bmatrix} + \end{equation*} + $$ + + Use this inverses to solve + + $$ + \begin{equation*} + A_0 x = \begin{bmatrix} + 3\\ + 1 + \end{bmatrix} + \end{equation*} + $$ + $$ + \begin{equation*} + A_0 x = \begin{bmatrix} + 2\\ + -5 + \end{bmatrix} + \end{equation*} + $$ + + Note: it is not common to solve $Ax=b$ using matrix inversion.\\ + + Reasons: + \begin{itemize} + \item $Ax=b$ can be solvable when $A$ is not invertible. + \item It is often slower / more costly + \end{itemize} + + \solution + + \part + + \begin{multicols}{2} + + $$ + \begin{align*} + A^{-1} &\implies \left[ \begin{array}{rr|rr} + -2&-3&1&0\\ + -6&-4&0&1 + \end{array}\right]\\ + -\frac{1}{2}\rho_1, -\frac{1}{4}\rho_2 &\implies \left[ \begin{array}{rr|rr} + 1&\frac{3}{2}&-\frac{1}{2}&0\\ + \frac{3}{2}&1&0&-\frac{1}{4} + \end{array}\right]\\ + \frac{3}{2}\rho_1 - \rho_2 &\implies \left[ \begin{array}{rr|rr} + 1&\frac{3}{2}&-\frac{1}{2}&0\\ + 0&\frac{5}{4}&-\frac{3}{4}&\frac{1}{4} + \end{array}\right]\\ + \frac{4}{5} \rho_2 &\implies \left[ \begin{array}{rr|rr} + 1&\frac{3}{2}&-\frac{1}{2}&0\\ + 0&1&-\frac{3}{5}&\frac{1}{5} + \end{array}\right]\\ + \rho_1 - \frac{3}{2}\rho_2 &\implies \left[ \begin{array}{rr|rr} + 1&0&\frac{2}{5}&-\frac{3}{10}\\ + 0&1&-\frac{3}{5}&\frac{1}{5} + \end{array}\right]\\\\ + A^{-1} &= \begin{bmatrix} + \frac{2}{5}&-\frac{3}{10}\\ + \frac{3}{5}&\frac{1}{10} + \end{bmatrix}\\ + \end{align*} + $$ + + \columnbreak + + $$ + \begin{align*} + A_0 x = \begin{bmatrix} + 3\\ + 1 + \end{bmatrix}\\ + A_0^{-1} A_0 x &= A_0^{-1} \begin{bmatrix} + 3\\ + 1 + \end{bmatrix}\\ + x &= \begin{bmatrix} + \frac{2}{5}\\ + \frac{3}{5} + \end{bmatrix} + \begin{bmatrix} + 3\\ + 1 + \end{bmatrix}\\ + &= \begin{bmatrix} + 0.9\\ + -1.6 + \end{bmatrix} + \end{align*} + $$ + + \end{multicols} + + \pagebreak + + + \part + + \begin{multicols}{2} + + $$ + \begin{align*} + A^{-1} &\implies \left[ \begin{array}{rr|rr} + 3&1&1&0\\ + -2&2&0&1 + \end{array}\right]\\ + \rho_2 + \rho_1 &\implies \left[ \begin{array}{rr|rr} + 3&1&1&0\\ + 1&3&1&1 + \end{array}\right]\\ + \frac{1}{3}\rho_1 &\implies \left[ \begin{array}{rr|rr} + 1&\frac{1}{3}&\frac{1}{3}&0\\ + 1&3&1&1 + \end{array}\right]\\ + \rho_1-\rho_2 &\implies \left[ \begin{array}{rr|rr} + 1&\frac{1}{3}&\frac{1}{3}&0\\ + 0&1&\frac{1}{4}&\frac{3}{8} + \end{array}\right]\\ + -\frac{3}{8}\rho_2 &\implies \left[ \begin{array}{rr|rr} + 1&\frac{1}{3}&\frac{1}{3}&0\\ + 0&1&\frac{1}{4}&\frac{3}{8} + \end{array}\right]\\ + \rho_1 - \frac{1}{3}\rho_2 &\implies \left[ \begin{array}{rr|rr} + 1&0&\frac{1}{4}&-\frac{1}{8}\\ + 0&1&\frac{1}{4}&\frac{3}{8} + \end{array}\right]\\ + A^{-1} &= \begin{bmatrix} + \frac{1}{4}&-\frac{1}{8}\\ + \frac{1}{4}&\frac{3}{8} + \end{bmatrix}\\ + \end{align*} + $$ + + \columnbreak + + $$ + \begin{align*} + A_0 x &= \begin{bmatrix} + 2\\ + -5 + \end{bmatrix}\\ + A_0^{-1} A_0 x &= A_0^{-1} \begin{bmatrix} + 2\\ + -5 + \end{bmatrix}\\ + x &= \begin{bmatrix} + \frac{1}{4}&-\frac{1}{8}\\ + \frac{1}{4}&\frac{3}{8} + \end{bmatrix} + \begin{bmatrix} + 2\\ + -5 + \end{bmatrix}\\ + &= \begin{bmatrix} + \frac{9}{8}\\ + -\frac{11}{8} + \end{bmatrix} + \end{align*} + $$ + + \end{multicols} + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Compute the determinant of + + $$ + \begin{equation*} + A = \begin{bmatrix} + 1&2&-1\\ + 2&0&2\\ + 4&3&1 + \end{bmatrix} + \end{equation*} + \begin{equation*} + A = \begin{bmatrix} + 2&-4&3\\ + 1&1&1\\ + 3&3&1 + \end{bmatrix} + \end{equation*} + \begin{equation*} + A = \begin{bmatrix} + 1&0&1\\ + 2&3&2\\ + -1&3&2 + \end{bmatrix} + \end{equation*} + \begin{equation*} + A = \begin{bmatrix} + 1&2&3\\ + -5&-10&-15\\ + 6&12&18 + \end{bmatrix} + \end{equation*} + $$ + + \begin{itemize} + \item Are they Invertible? + \item Which of them has full rank? Which of them has lower rank and which one? + \end{itemize} + + \solution + + \part + + \begin{multicols}{2} + + $$ + \begin{bmatrix} + 1&2&-1\\ + 2&0&2\\ + 4&3&1 + \end{bmatrix} + $$ + + $$ + \begin{align*} + \det(A) &= 0+16-6-(0+6+4)\\ + &= 0 + \end{align*} + $$ + + This matrix is not invertible. + + \columnbreak + + $$ + \begin{bmatrix} + 2&-4&3\\ + 1&1&1\\ + 3&3&1 + \end{bmatrix} + $$ + + $$ + \begin{align*} + \det(A) &= 2-1+9-9-6+4\\ + &= -12 + \end{align*} + $$ + + This matrix is invertible + + \end{multicols} + + + \begin{multicols}{2} + + $$ + \begin{bmatrix} + 1&0&1\\ + 2&3&2\\ + -1&3&2 + \end{bmatrix} + $$ + + $$ + \begin{align*} + \det(A) &= 6+0+6+3-6-0\\ + &= 9 + \end{align*} + $$ + + This matrix is invertible + + \columnbreak + + $$ + \begin{bmatrix} + 1&2&3\\ + -5&-10&-15\\ + 6&12&18 + \end{bmatrix} + $$ + + $$ + \begin{align*} + \det(A) &= -180-180-180+180+180+180\\ + &= 0 + \end{align*} + $$ + + This matrix is not invertible + + \end{multicols} + + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + For what value $a$ the matrix is not invertible? + + $$ + \begin{equation*} + A = \begin{bmatrix} + 1&2&1\\ + 2&a&4\\ + -3&1&2 + \end{bmatrix} + \end{equation*} + $$ + + \solution + + A matrix is not invertible if its determinant is zero. + + $$ + \begin{align*} + \det(A) &= 2a-24+2+3a-4-8\\ + &= 5a-34\\ + 5a-34 &= 0\\ + a &= \frac{34}{5} \approx 6.8 + \end{align*} + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Compute and apply the Householder matrix which makes transforms the first column of + to a multple of the first one-hot vector $\begin{bmatrix}1\\0\\0\end{bmatrix}$ for + + $$ + \begin{equation*} + A = \begin{bmatrix} + 1&2&-1\\ + 2&0&2\\ + 2&3&1 + \end{bmatrix} + \end{equation*} + $$ + + and for (hint: here subtracting is nicer) + + $$ + \begin{equation*} + A = \begin{bmatrix} + 1&-4&3\\ + 3&1&1\\ + \sqrt{6}&3&1 + \end{bmatrix} + \end{equation*} + $$ + + \solution + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Verify that + + $$ + \begin{equation*} + A = \begin{bmatrix} + 1&0&0\\ + 0&\cos(\alpha)&-\sin(\alpha)\\ + 0&\sin(\alpha)&\cos(\alpha) + \end{bmatrix} + \end{equation*} + $$ + + Satisfies being an orthogonal matrix. + + \solution + + $$ + \begin{align*} + \det(A) &= 0 + (-\sin^2(\alpha)) + 0 - \cos^2(\alpha) -0 -0\\ + &=-(sin^2(\alpha) + cos^2(\alpha))\\ + &= -1 + \end{align*} + $$ + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + + What is the cosine of the angle between + + $$ + \left( 6,-6,-4,\sqrt{12} \right), \left( 6,4,2,\sqrt{25} \right)? + $$ + + \solution + + $$ + \begin{align*} + \cos(\angle(\vec{a},\vec{b})) &= \frac{\vec{a}\cdot\vec{b}}{\|\vec{a}\|\|\vec{b}\|}\\ + &= \frac{6\cdot6 + (-6)\cdot4 + (-4)\cdot2 + \sqrt{12}\cdot\sqrt{25}}{\sqrt{6^2+(-6)^2+(-4)^2+\sqrt{12}^2}\sqrt{6^2+4^2+2^2+\sqrt{25}^2}}\\ + &= \frac{36-24-8+5\sqrt{12}}{\sqrt{100}\sqrt{81}}\\ + &= \frac{4}{90}\\ + \end{align*} +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Another $3\times3$ affine system + + \begin{itemize} + \item show the intermediate result when the first column is the one hot vector $\begin{bmatrix}1&0&0\end{bmatrix}$ for the first time + \item show that the intermediate result when the matrix has row echelon form for the first time + \item get the solution + \end{itemize} + + $$ + \begin{align*} + 2x-3y+2z &=-4\\ + 7x+4.5y-1z &=16\\ + 4x+3y+z &=2 + \end{align*} + $$ + + \solution + + \begin{multicols}{2} + + $$ + \begin{align*} + &\left[ \begin{array}{rrr|r} + 2&-3&2&-4\\ + 7&4.5&-1&16\\ + 4&3&1&2 + \end{array} \right]\\ + \frac{1}{2} \rho_1 &\implies \left[ \begin{array}{rrr|r} + 1&-3/2&1&-2\\ + 7&4.5&-1&16\\ + 4&3&1&2 + \end{array} \right]\\ + (i) \rho_2 - 7\rho_1, \rho_3 - 4\rho_1 &\implies \left[ \begin{array}{rrr|r} + 1&-3/2&1&-2\\ + 0&15&-8&30\\ + 0&9&-3&10 + \end{array} \right]\\ + \frac{1}{15} \rho_2, -\frac{1}{3} \rho_3 &\implies \left[ \begin{array}{rrr|r} + 1&-3/2&1&-2\\ + 0&1&-8/15&2\\ + 0&-3&1&-\frac{10}{3} + \end{array} \right]\\ + (ii) \rho_3 + 3\rho_2 &\implies \left[ \begin{array}{rrr|r} + 1&-3/2&1&-2\\ + 0&1&-8/15&2\\ + 0&0&1&-\frac{40}{9} + \end{array} \right]\\ + \rho_2 + \frac{8}{15}\rho_3 &\implies \left[ \begin{array}{rrr|r} + 1&-3/2&1&-2\\ + 0&1&0&-\frac{10}{27}\\ + 0&0&1&-\frac{40}{9} + \end{array} \right]\\ + \rho_1+\frac{3}{2}\rho_2 - \rho_3 &\implies \left[ \begin{array}{rrr|r} + 1&0&0&-\frac{17}{9}\\ + 0&1&0&-\frac{10}{27}\\ + 0&0&1&-\frac{40}{9} + \end{array} \right]\\ + \begin{bmatrix} + x\\ + y\\ + z + \end{bmatrix} &= \begin{bmatrix} + -\frac{17}{9}\\ + -\frac{10}{27}\\ + -\frac{40}{9} + \end{bmatrix} + \end{align*} + $$ + \end{multicols} + +\end{homeworkProblem} + + +\end{document} diff --git a/Tutorial 10/INF1004_W10_Tutorial_2200624_Solution.synctex.gz b/Tutorial 10/INF1004_W10_Tutorial_2200624_Solution.synctex.gz new file mode 100644 index 0000000000000000000000000000000000000000..93b179531f62144f9a0cfd8a0c5233dd2d7bfca5 GIT 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\enterProblemHeader{homeworkProblemCounter} +}{ + \exitProblemHeader{homeworkProblemCounter} +} + +\newcommand{\hmwkClass}{INF 1004 Mathematics 2} +\newcommand{\hmwkTitle}{Tutorial\ \#10\ Solutions} +\newcommand{\hmwkAuthorName}{\textbf{Woon Jun Wei}} +\newcommand{\hmwkStudentID}{\textbf{2200624}} + +\title{ + \vspace{2in} + \textmd{\textbf{\hmwkClass \\ \hmwkTitle}}\\ + \vspace{3in} +} + +\author{\hmwkAuthorName \\ \hmwkStudentID} +\date{\today} + +\renewcommand{\part}[1]{\textbf{\large Part \arabic{partCounter}}\stepcounter{partCounter}\\} + +% +% Various Helper Commands +% + +% Useful for algorithms +\newcommand{\alg}[1]{\textsc{\bfseries \footnotesize #1}} + +% For derivatives +\newcommand{\deriv}[1]{\frac{\mathrm{d}}{\mathrm{d}x} (#1)} + +% For partial derivatives +\newcommand{\pderiv}[2]{\frac{\partial}{\partial #1} (#2)} + +% Integral dx +\newcommand{\dx}{\mathrm{d}x} + +% Alias for the Solution section header +\newcommand{\solution}{\textbf{\large My Solution\\}} +\newcommand{\answer}{\textbf{\large Sample Solutions\\}} + +% Probability commands: Expectation, Variance, Covariance, Bias +\newcommand{\E}{\mathrm{E}} +\newcommand{\Var}{\mathrm{Var}} +\newcommand{\Cov}{\mathrm{Cov}} +\newcommand{\Bias}{\mathrm{Bias}} + +\begin{document} + +\maketitle + +\pagebreak + +\begin{homeworkProblem} + Compute $A^\top A$ for + $$ + \begin{equation*} + A = \begin{bmatrix} + 2&-3\\ + -6&-9 + \end{bmatrix} + \end{equation*} + + \begin{equation*} + A = \begin{bmatrix} + 0&1&-1\\ + 2&0&2\\ + 4&3&1 + \end{bmatrix} + \end{equation*} + $$ + + \solution + + \part + + $$ + \begin{align*} + A &= \begin{bmatrix} + 2&-3\\ + -6&-9 + \end{bmatrix}\\ + A^\top &= \begin{bmatrix} + 2&-6\\ + -3&-9 + \end{bmatrix}\\ + A^\top A &= \begin{bmatrix} + 2&-6\\ + -3&-9 + \end{bmatrix} + \begin{bmatrix} + 2&-3\\ + -6&-9 + \end{bmatrix}\\ + &= \begin{bmatrix} + 40&48\\ + 48&100 + \end{bmatrix} + \end{align*} + $$ + + \part + + $$ + \begin{align*} + A &= \begin{bmatrix} + 0&1&-1\\ + 2&0&2\\ + 4&3&1 + \end{bmatrix}\\ + A^\top &= \begin{bmatrix} + 0&2&4\\ + 1&0&3\\ + -1&2&1 + \end{bmatrix}\\ + A^\top A &= \begin{bmatrix} + 0&2&4\\ + 1&0&3\\ + -1&2&1 + \end{bmatrix} + \begin{bmatrix} + 0&1&-1\\ + 2&0&2\\ + 4&3&1 + \end{bmatrix}\\ + &= \begin{bmatrix} + 20&12&8\\ + 12&10&2\\ + 8&2&6 + \end{bmatrix} + \end{align*} + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Compute the inverse of + $$ + \begin{equation*} + A_0 = \begin{bmatrix} + -2&-3\\ + -6&-4 + \end{bmatrix} + \end{equation*} + $$ + $$ + \begin{equation*} + A_1 = \begin{bmatrix} + 3&1\\ + -2&2 + \end{bmatrix} + \end{equation*} + $$ + + Use this inverses to solve + + $$ + \begin{equation*} + A_0 x = \begin{bmatrix} + 3\\ + 1 + \end{bmatrix} + \end{equation*} + $$ + $$ + \begin{equation*} + A_0 x = \begin{bmatrix} + 2\\ + -5 + \end{bmatrix} + \end{equation*} + $$ + + Note: it is not common to solve $Ax=b$ using matrix inversion.\\ + + Reasons: + \begin{itemize} + \item $Ax=b$ can be solvable when $A$ is not invertible. + \item It is often slower / more costly + \end{itemize} + + \solution + + \part + + \begin{multicols}{2} + + $$ + \begin{align*} + A^{-1} &\implies \left[ \begin{array}{rr|rr} + -2&-3&1&0\\ + -6&-4&0&1 + \end{array}\right]\\ + -\frac{1}{2}\rho_1, -\frac{1}{4}\rho_2 &\implies \left[ \begin{array}{rr|rr} + 1&\frac{3}{2}&-\frac{1}{2}&0\\ + \frac{3}{2}&1&0&-\frac{1}{4} + \end{array}\right]\\ + \frac{3}{2}\rho_1 - \rho_2 &\implies \left[ \begin{array}{rr|rr} + 1&\frac{3}{2}&-\frac{1}{2}&0\\ + 0&\frac{5}{4}&-\frac{3}{4}&\frac{1}{4} + \end{array}\right]\\ + \frac{4}{5} \rho_2 &\implies \left[ \begin{array}{rr|rr} + 1&\frac{3}{2}&-\frac{1}{2}&0\\ + 0&1&-\frac{3}{5}&\frac{1}{5} + \end{array}\right]\\ + \rho_1 - \frac{3}{2}\rho_2 &\implies \left[ \begin{array}{rr|rr} + 1&0&\frac{2}{5}&-\frac{3}{10}\\ + 0&1&-\frac{3}{5}&\frac{1}{5} + \end{array}\right]\\\\ + A^{-1} &= \begin{bmatrix} + \frac{2}{5}&-\frac{3}{10}\\ + \frac{3}{5}&\frac{1}{10} + \end{bmatrix}\\ + \end{align*} + $$ + + \columnbreak + + $$ + \begin{align*} + A_0 x = \begin{bmatrix} + 3\\ + 1 + \end{bmatrix}\\ + A_0^{-1} A_0 x &= A_0^{-1} \begin{bmatrix} + 3\\ + 1 + \end{bmatrix}\\ + x &= \begin{bmatrix} + \frac{2}{5}\\ + \frac{3}{5} + \end{bmatrix} + \begin{bmatrix} + 3\\ + 1 + \end{bmatrix}\\ + &= \begin{bmatrix} + 0.9\\ + -1.6 + \end{bmatrix} + \end{align*} + $$ + + \end{multicols} + + \pagebreak + + + \part + + \begin{multicols}{2} + + $$ + \begin{align*} + A^{-1} &\implies \left[ \begin{array}{rr|rr} + 3&1&1&0\\ + -2&2&0&1 + \end{array}\right]\\ + \rho_2 + \rho_1 &\implies \left[ \begin{array}{rr|rr} + 3&1&1&0\\ + 1&3&1&1 + \end{array}\right]\\ + \frac{1}{3}\rho_1 &\implies \left[ \begin{array}{rr|rr} + 1&\frac{1}{3}&\frac{1}{3}&0\\ + 1&3&1&1 + \end{array}\right]\\ + \rho_1-\rho_2 &\implies \left[ \begin{array}{rr|rr} + 1&\frac{1}{3}&\frac{1}{3}&0\\ + 0&1&\frac{1}{4}&\frac{3}{8} + \end{array}\right]\\ + -\frac{3}{8}\rho_2 &\implies \left[ \begin{array}{rr|rr} + 1&\frac{1}{3}&\frac{1}{3}&0\\ + 0&1&\frac{1}{4}&\frac{3}{8} + \end{array}\right]\\ + \rho_1 - \frac{1}{3}\rho_2 &\implies \left[ \begin{array}{rr|rr} + 1&0&\frac{1}{4}&-\frac{1}{8}\\ + 0&1&\frac{1}{4}&\frac{3}{8} + \end{array}\right]\\ + A^{-1} &= \begin{bmatrix} + \frac{1}{4}&-\frac{1}{8}\\ + \frac{1}{4}&\frac{3}{8} + \end{bmatrix}\\ + \end{align*} + $$ + + \columnbreak + + $$ + \begin{align*} + A_0 x &= \begin{bmatrix} + 2\\ + -5 + \end{bmatrix}\\ + A_0^{-1} A_0 x &= A_0^{-1} \begin{bmatrix} + 2\\ + -5 + \end{bmatrix}\\ + x &= \begin{bmatrix} + \frac{1}{4}&-\frac{1}{8}\\ + \frac{1}{4}&\frac{3}{8} + \end{bmatrix} + \begin{bmatrix} + 2\\ + -5 + \end{bmatrix}\\ + &= \begin{bmatrix} + \frac{9}{8}\\ + -\frac{11}{8} + \end{bmatrix} + \end{align*} + $$ + + \end{multicols} + + \answer + + \begin{multicols}{2} + \textbf{A:}\\ + + $$ + \begin{align*} + \det(A_0) &= (-2)(-4)-(-3)(-6) = -10\\ + A_0^{-1} &= \frac{1}{\det(A_0)} \begin{bmatrix} + -4&3\\ + 6&-2 + \end{bmatrix}\\ + x &= A_0^{-1} \begin{bmatrix} + 3\\ + 1 + \end{bmatrix}\\ + &= -\frac{1}{10} \begin{bmatrix} + -4&3\\ + 6&-2 + \end{bmatrix} + \begin{bmatrix} + 3\\ + 1 + \end{bmatrix}\\ + &= \begin{bmatrix} + 0.9\\ + -1.6 + \end{bmatrix} + \end{align*} + $$ + + \columnbreak + + \textbf{B:}\\ + + $$ + \begin{align*} + \det{A_1} &= 3(2) - (-2)(1) = 8\\ + A_1^{-1} &= \frac{1}{8} \begin{bmatrix} + 2&-1\\ + 2&3 + \end{bmatrix}\\ + x &= \frac{1}{8} \begin{bmatrix} + 2&-1\\ + 2&3 + \end{bmatrix} + \begin{bmatrix} + 2\\ + -5 + \end{bmatrix}\\ + &= \begin{bmatrix} + \frac{9}{8}\\ + -\frac{11}{8} + \end{bmatrix} + \end{align*} + $$ + + \end{multicols} + + \textbf{FORMULA:}\\ + + $$ + A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} + d&-b\\-c&a + \end{bmatrix} + $$ + + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Compute the determinant of + + $$ + \begin{equation*} + A = \begin{bmatrix} + 1&2&-1\\ + 2&0&2\\ + 4&3&1 + \end{bmatrix} + \end{equation*} + \begin{equation*} + A = \begin{bmatrix} + 2&-4&3\\ + 1&1&1\\ + 3&3&1 + \end{bmatrix} + \end{equation*} + \begin{equation*} + A = \begin{bmatrix} + 1&0&1\\ + 2&3&2\\ + -1&3&2 + \end{bmatrix} + \end{equation*} + \begin{equation*} + A = \begin{bmatrix} + 1&2&3\\ + -5&-10&-15\\ + 6&12&18 + \end{bmatrix} + \end{equation*} + $$ + + \begin{itemize} + \item Are they Invertible? + \item Which of them has full rank? Which of them has lower rank and which one? + \end{itemize} + + \textbf{Notes:} \\ + + \begin{itemize} + \item Rank is the number of linearly independent rows or columns. + \item A matrix is invertible if they have full rank. + \end{itemize} + + \answer + + + \begin{multicols}{2} + + \part + + $$ + \begin{bmatrix} + 1&2&-1\\ + 2&0&2\\ + 4&3&1 + \end{bmatrix} \rightarrow \begin{bmatrix} + 1&2&-1\\ + 0&1&-1\\ + 0&0&0 + \end{bmatrix} + $$ + + $$ + \begin{align*} + \det(A) &= 0+16-6-(0+6+4)\\ + &= 0 + \end{align*} + $$ + + This matrix is not invertible. Rank is 2. + + \columnbreak + + \part + + $$ + \begin{bmatrix} + 2&-4&3\\ + 1&1&1\\ + 3&3&1 + \end{bmatrix} \rightarrow \begin{bmatrix} + 1&1&-1\\0&1&-\frac{5}{6}\\0&0&1 + \end{bmatrix} + + $$ + + $$ + \begin{align*} + \det(A) &= 2-1+9-9-6+4\\ + &= -12 + \end{align*} + $$ + + This matrix is invertible. Rank is 3. + + \end{multicols} + + \pagebreak + + \part + + \begin{multicols}{2} + + $$ + \begin{bmatrix} + 1&0&1\\ + 2&3&2\\ + -1&3&2 + \end{bmatrix} \rightarrow \begin{bmatrix} + 1&0&1\\ + 0&1&1\\ + 0&0&1 + \end{bmatrix} + $$ + + $$ + \begin{align*} + \det(A) &= 6+0+6+3-6-0\\ + &= 9 + \end{align*} + $$ + + This matrix is invertible. Rank is 3. + + \columnbreak + + \part + + $$ + \begin{bmatrix} + 1&2&3\\ + -5&-10&-15\\ + 6&12&18 + \end{bmatrix} \rightarrow \begin{bmatrix} + 1&2&3\\ + 0&0&0\\ + 0&0&0 + \end{bmatrix} + $$ + + $$ + \begin{align*} + \det(A) &= -180-180-180+180+180+180\\ + &= 0 + \end{align*} + $$ + + This matrix is not invertible and has rank 1. + + \end{multicols} + + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + For what value $a$ the matrix is not invertible? + + $$ + \begin{equation*} + A = \begin{bmatrix} + 1&2&1\\ + 2&a&4\\ + -3&1&2 + \end{bmatrix} + \end{equation*} + $$ + + \solution + + A matrix is not invertible if its determinant is zero. + + $$ + \begin{align*} + \det(A) &= 2a-24+2+3a-4-8\\ + &= 5a-34\\ + 5a-34 &= 0\\ + a &= \frac{34}{5} \approx 6.8 + \end{align*} + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Compute and apply the Householder matrix which makes transforms the first column of + to a multple of the first one-hot vector $\begin{bmatrix}1\\0\\0\end{bmatrix}$ for + + $$ + \begin{equation*} + A = \begin{bmatrix} + 1&2&-1\\ + 2&0&2\\ + 2&3&1 + \end{bmatrix} + \end{equation*} + $$ + + and for (hint: here subtracting is nicer) + + $$ + \begin{equation*} + A = \begin{bmatrix} + 1&-4&3\\ + 3&1&1\\ + \sqrt{6}&3&1 + \end{bmatrix} + \end{equation*} + $$ + + \textbf{Formula:} \\ + $$ + \begin{align*} + u &= x \pm \left\lVert x \right\rVert_2 z\\ + H_u &= I - \frac{2}{u \cdot u} uu^T\\ + \text{Properties of H: }\\ + H_ux &= \mp \left\lVert x \right\rVert_2 z \text{ H maps x to $\mp \left\lvert x \right\rvert z$ given $u = x \pm \lvert x \rvert z$}\\ + H_u^\top H_u &= I \text{ H is an orthogonal matrix}\\ + H_u &= H_u^\top \text{ H is a symmetric matrix}\\ + \end{align*} + $$ + + \textbf{Example:}\\ + + Here $A = \begin{bmatrix}3&1\\4&4\end{bmatrix}$ + + $x = \begin{bmatrix}3\\4\end{bmatrix}$ + + $\right \lVert x \right \rVert_2 = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$ + + \begin{multicols}{2} + + $$ + \begin{align*} + u &= \begin{bmatrix}3\\4\end{bmatrix} \pm 5 \begin{bmatrix}1\\0\end{bmatrix} = \begin{bmatrix}-2\\4\end{bmatrix}\text{ (Choose Minus)}\\ + H &= \begin{bmatrix}1&0\\0&1\end{bmatrix}-\frac{2}{(-2)^2+4^2}\begin{bmatrix}-2\\4\end{bmatrix}\begin{bmatrix}-2&4\end{bmatrix}\\ + &= \begin{bmatrix}1&0\\0&1\end{bmatrix}-\frac{2}{20}\begin{bmatrix}4&-8\\-8&16\end{bmatrix}\\ + &= \begin{bmatrix}0.6&0.8\\0.8&-0.6\end{bmatrix}\\\\ + HA &= \begin{bmatrix}0.6&0.8\\0.8&-0.6\end{bmatrix}\begin{bmatrix}3&1\\4&4\end{bmatrix}\\ + &= \begin{bmatrix}5&3.8\\0&-1.6\end{bmatrix} + \end{align*} + $$ + + \columnbreak + + $$ + \begin{align*} + u &= \begin{bmatrix}3\\4\end{bmatrix} \pm 5 \begin{bmatrix}1\\0\end{bmatrix} = \begin{bmatrix}8\\4\end{bmatrix}\text{ (Choose Positive Sign)}\\ + H &= \begin{bmatrix}1&0\\0&1\end{bmatrix}-\frac{2}{(8)^2+4^2}\begin{bmatrix}8\\4\end{bmatrix}\begin{bmatrix}8&4\end{bmatrix}\\ + &= \begin{bmatrix}1&0\\0&1\end{bmatrix}-\frac{2}{80}\begin{bmatrix}64&32\\32&16\end{bmatrix}\\ + &= \begin{bmatrix}-0.6&-0.8\\-0.8&0.6\end{bmatrix}\\\\ + HA &= \begin{bmatrix}-0.6&-0.8\\-0.8&0.6\end{bmatrix}\begin{bmatrix}3&1\\4&4\end{bmatrix}\\ + &= \begin{bmatrix}-5&-3.8\\0&-1.6\end{bmatrix} + \end{align*} + $$ + + \end{multicols} + + \pagebreak + + \part + + \begin{multicols}{2} + + $$ + \begin{align*} + u &= \begin{bmatrix}1\\2\\2\end{bmatrix} \pm \lVert \begin{pmatrix}1,2,2\end{pmatrix} \rVert \begin{bmatrix}1\\0\\0\end{bmatrix} \\ + &= \begin{bmatrix}1\\2\\2\end{bmatrix} \pm \sqrt{1^2 + 2^2 + 2^2} \begin{bmatrix}1\\0\\0\end{bmatrix}\\ + &= \begin{bmatrix}1\\2\\2\end{bmatrix} - 3 \begin{bmatrix}1\\0\\0\end{bmatrix}\\ + &= \begin{bmatrix}1\\2\\2\end{bmatrix} - \begin{bmatrix}3\\0\\0\end{bmatrix}\\ + &= \begin{bmatrix}-2\\2\\2\end{bmatrix}\\ + H &= I - \frac{2}{12}uu^\top = I - \frac{1}{6}\begin{bmatrix}-2\\2\\2\end{bmatrix} \begin{bmatrix}-2&2&2\end{bmatrix} \\ + &= I - \frac{1}{6}\begin{bmatrix}4&-4&-4\\-4&4&4\\-4&4&4\end{bmatrix}\\ + &= \begin{bmatrix} 1&0&0\\0&1&0\\0&0&1\end{bmatrix} - \frac{1}{6}\begin{bmatrix}4&-4&-4\\-4&4&4\\-4&4&4\end{bmatrix}\\ + &= \begin{bmatrix}\frac{1}{3}&\frac{2}{3}&\frac{2}{3}\\\frac{2}{3}&\frac{1}{3}&\frac{1}{3}\\ \frac{2}{3}&\frac{1}{3}&\frac{1}{3}\end{bmatrix}\\ + HA &= \begin{bmatrix}3&2.67&1.67\\0&-0.67&-0.67\\0&2.33&-1.67\end{bmatrix} + \end{align*} + $$ + + \columnbreak + + \part + + $$ + \begin{align*} + u &= x - \lVert x \lVert z = \begin{bmatrix}1\\3\\\sqrt{6}\end{bmatrix} - 4\begin{bmatrix} 1\\0\\0 \end{bmatrix} = \begin{bmatrix} -3\\3\\\sqrt{6}\end{bmatrix}\\ + H &= I - \frac{2}{u\cdot u} uu^\top\\ + &= I - \frac{2}{24}\begin{bmatrix}-3\\3\\\sqrt{6}\end{bmatrix}\begin{bmatrix}-3&3&\sqrt{6}\end{bmatrix}\\ + &= \begin{bmatrix}\frac{1}{4}&\frac{3}{4}&\frac{\sqrt{6}}{4}\\\frac{3}{4}&\frac{1}{4}&-\frac{\sqrt{6}}{4}\\\frac{\sqrt{6}}{4}&-\frac{\sqrt{6}}{4}&\frac{1}{2}\end{bmatrix}\\ + HA &= \begin{bmatrix} 4&*&*\\0&*&*\\0&*&*\end{bmatrix}\\\\ + u &= x + \lVert x \lVert z = \begin{bmatrix}1\\3\\\sqrt{6}\end{bmatrix} + 4\begin{bmatrix} 1\\0\\0 \end{bmatrix} = \begin{bmatrix} 5\\3\\\sqrt{6}\end{bmatrix}\\ + H &= I - \frac{2}{u\cdot u} uu^\top\\ + &= \begin{bmatrix}-\frac{1}{4}&-\frac{3}{4}&-\frac{\sqrt{6}}{4}\\-\frac{3}{4}&\frac{11}{20}&-\frac{3\sqrt{6}}{70}\\-\frac{\sqrt{6}}{4}&-\frac{3\sqrt{6}}{70}&\frac{7}{10}\end{bmatrix}\\ + HA &= \begin{bmatrix} -4&*&*\\0&*&*\\0&*&*\end{bmatrix} + \end{align*} + $$ + + +\end{multicols} + + + + + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Verify that + + $$ + \begin{equation*} + A = \begin{bmatrix} + 1&0&0\\ + 0&\cos(\alpha)&-\sin(\alpha)\\ + 0&\sin(\alpha)&\cos(\alpha) + \end{bmatrix} + \end{equation*} + $$ + + Satisfies being an orthogonal matrix. + + Note: Cannot use determinant to verify. + + \answer + + $$ + \begin{align*} + A^\top A &= \begin{bmatrix} + 1&0&0\\ + 0&\cos(\alpha)&\sin(\alpha)\\ + 0&-\sin(\alpha)&\cos(\alpha) + \end{bmatrix} \begin{bmatrix} + 1&0&0\\ + 0&\cos(\alpha)&-\sin(\alpha)\\ + 0&\sin(\alpha)&\cos(\alpha) + \end{bmatrix}\\ + &= \begin{bmatrix} + 1&0&0\\ + 0&\cos^2(\alpha) + \sin^2(\alpha)&-\sin(\alpha)\cos(\alpha) + \sin(\alpha)\cos(\alpha)\\ + 0&-\sin(\alpha)\cos(\alpha) + \sin(\alpha)\cos(\alpha)&\sin^2(\alpha) + \cos^2(\alpha) + \end{bmatrix}\\ + &= \begin{bmatrix} + 1&0&0\\ + 0&1&0\\ + 0&0&1 + \end{bmatrix}\\ + \end{align*} + $$ + Thus A is an orthogonal matrix. + +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + + What is the cosine of the angle between + + $$ + \left( 6,-6,-4,\sqrt{12} \right), \left( 6,4,2,\sqrt{25} \right)? + $$ + + \solution + + $$ + \begin{align*} + \cos(\angle(\vec{a},\vec{b})) &= \frac{\vec{a}\cdot\vec{b}}{\|\vec{a}\|\|\vec{b}\|}\\ + &= \frac{6\cdot6 + (-6)\cdot4 + (-4)\cdot2 + \sqrt{12}\cdot\sqrt{25}}{\sqrt{6^2+(-6)^2+(-4)^2+\sqrt{12}^2}\sqrt{6^2+4^2+2^2+\sqrt{25}^2}}\\ + &= \frac{36-24-8+5\sqrt{12}}{\sqrt{100}\sqrt{81}}\\ + &= \frac{4 + 5\sqrt{12}}{90}\\ + $\approx 0.24 + \end{align*} +\end{homeworkProblem} + +\pagebreak + +\begin{homeworkProblem} + Another $3\times3$ affine system + + \begin{itemize} + \item show the intermediate result when the first column is the one hot vector $\begin{bmatrix}1&0&0\end{bmatrix}$ for the first time + \item show that the intermediate result when the matrix has row echelon form for the first time + \item get the solution + \end{itemize} + + $$ + \begin{align*} + 2x-3y+2z &=-4\\ + 7x+4.5y-1z &=16\\ + 4x+3y+z &=2 + \end{align*} + $$ + + \solution + + \begin{multicols}{2} + + $$ + \begin{align*} + &\left[ \begin{array}{rrr|r} + 2&-3&2&-4\\ + 7&4.5&-1&16\\ + 4&3&1&2 + \end{array} \right]\\ + \frac{1}{2} \rho_1 &\implies \left[ \begin{array}{rrr|r} + 1&-3/2&1&-2\\ + 7&4.5&-1&16\\ + 4&3&1&2 + \end{array} \right]\\ + (i) \rho_2 - 7\rho_1, \rho_3 - 4\rho_1 &\implies \left[ \begin{array}{rrr|r} + 1&-3/2&1&-2\\ + 0&15&-8&30\\ + 0&9&-3&10 + \end{array} \right]\\ + \frac{1}{15} \rho_2, -\frac{1}{3} \rho_3 &\implies \left[ \begin{array}{rrr|r} + 1&-3/2&1&-2\\ + 0&1&-8/15&2\\ + 0&-3&1&-\frac{10}{3} + \end{array} \right]\\ + (ii) \rho_3 + 3\rho_2 &\implies \left[ \begin{array}{rrr|r} + 1&-3/2&1&-2\\ + 0&1&-8/15&2\\ + 0&0&1&-\frac{40}{9} + \end{array} \right]\\ + \rho_2 + \frac{8}{15}\rho_3 &\implies \left[ \begin{array}{rrr|r} + 1&-3/2&1&-2\\ + 0&1&0&-\frac{10}{27}\\ + 0&0&1&-\frac{40}{9} + \end{array} \right]\\ + \rho_1+\frac{3}{2}\rho_2 - \rho_3 &\implies \left[ \begin{array}{rrr|r} + 1&0&0&-\frac{17}{9}\\ + 0&1&0&-\frac{10}{27}\\ + 0&0&1&-\frac{40}{9} + \end{array} \right]\\ + \begin{bmatrix} + x\\ + y\\ + z + \end{bmatrix} &= \begin{bmatrix} + -\frac{17}{9}\\ + -\frac{10}{27}\\ + -\frac{40}{9} + \end{bmatrix} + \end{align*} + $$ + \end{multicols} + +\end{homeworkProblem} + + +\end{document} diff --git a/Tutorial 10/INF1004_W11_Tutorial_2200624_Solution.pdf b/Tutorial 10/INF1004_W11_Tutorial_2200624_Solution.pdf new file mode 100644 index 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mode 100644 index 0000000..8659621 --- /dev/null +++ b/Tutorial 11/INF1004_W11_Tutorial_2200624_Bare.tex @@ -0,0 +1,246 @@ +\documentclass[a4paper]{article} + +\usepackage{fancyhdr} +\usepackage{extramarks} +\usepackage{amsmath} +\usepackage{amsthm} +\usepackage{amsfonts} +\usepackage{tikz} +\usepackage{pgfplots} +\usepackage[plain]{algorithm} +\usepackage{algpseudocode} +\usepackage{enumitem} +\usepackage{framed} +\usepackage{multicol} + +\graphicspath{ {./images/} } + +\usetikzlibrary{automata,positioning} + +% +% Basic Document Settings +% + +\topmargin=-0.45in +\evensidemargin=0in +\oddsidemargin=0in +\textwidth=6.5in +\textheight=9.0in +\headsep=0.25in + +\linespread{1.1} + +\pagestyle{fancy} +\lhead{\hmwkAuthorName} +\chead{\hmwkClass\ : \hmwkTitle} +\rhead{\firstxmark} +\lfoot{\lastxmark} +\cfoot{\thepage} + +\renewcommand\headrulewidth{0.4pt} +\renewcommand\footrulewidth{0.4pt} + +\setlength\parindent{0pt} + +% +% Create Problem Sections +% + +\newcommand{\enterProblemHeader}[1]{ + \nobreak\extramarks{}{Question \arabic{#1} continued on next page\ldots}\nobreak{} + \nobreak\extramarks{Question \arabic{#1} (continued)}{Question \arabic{#1} continued on next page\ldots}\nobreak{} +} + +\newcommand{\exitProblemHeader}[1]{ + \nobreak\extramarks{Question \arabic{#1} (continued)}{Question \arabic{#1} continued on next page\ldots}\nobreak{} + \stepcounter{#1} + \nobreak\extramarks{Question \arabic{#1}}{}\nobreak{} +} + +\setcounter{secnumdepth}{0} +\newcounter{partCounter} +\newcounter{homeworkProblemCounter} +\setcounter{homeworkProblemCounter}{1} +\nobreak\extramarks{Question \arabic{homeworkProblemCounter}}{}\nobreak{} + +\newenvironment{homeworkProblem}[1][-1]{ + \ifnum#1>0 + \setcounter{homeworkProblemCounter}{#1} + \fi + \section{Question \arabic{homeworkProblemCounter}} + \setcounter{partCounter}{1} + \enterProblemHeader{homeworkProblemCounter} +}{ + \exitProblemHeader{homeworkProblemCounter} +} + +\newcommand{\hmwkClass}{INF 1004 Mathematics 2} +\newcommand{\hmwkTitle}{Tutorial\ \#11} +\newcommand{\hmwkAuthorName}{\textbf{Woon Jun Wei}} +\newcommand{\hmwkStudentID}{\textbf{2200624}} + +\title{ + \vspace{2in} + \textmd{\textbf{\hmwkClass \\ \hmwkTitle}}\\ + \vspace{3in} +} + +\author{\hmwkAuthorName \\ \hmwkStudentID} +\date{\today} + +\renewcommand{\part}[1]{\textbf{\large Part \arabic{partCounter}}\stepcounter{partCounter}\\} + +% +% Various Helper Commands +% + +% Useful for algorithms +\newcommand{\alg}[1]{\textsc{\bfseries \footnotesize #1}} + +% For derivatives +\newcommand{\deriv}[1]{\frac{\mathrm{d}}{\mathrm{d}x} (#1)} + +% For partial derivatives +\newcommand{\pderiv}[2]{\frac{\partial}{\partial #1} (#2)} + +% Integral dx +\newcommand{\dx}{\mathrm{d}x} + +% Alias for the Solution section header +\newcommand{\solution}{\textbf{\large My Solution\\}} +\newcommand{\answer}{\textbf{\large Sample Solutions\\}} + +% Probability commands: Expectation, Variance, Covariance, Bias +\newcommand{\E}{\mathrm{E}} +\newcommand{\Var}{\mathrm{Var}} +\newcommand{\Cov}{\mathrm{Cov}} +\newcommand{\Bias}{\mathrm{Bias}} + +\begin{document} + +\maketitle + +\pagebreak + +\begin{homeworkProblem} + \begin{itemize} + \item Compute the eigenvalues and eigenvectors for the matrices below. + \item For one of these matrices compute the matrix $P$ such that $P^{-1}DP = A$, and verify that $PAP^{-1}$ is the diagonal matrix of the eigenvalues + \end{itemize} + + $$ + \begin{align*} + A &= \begin{bmatrix} 4&\sqrt{15}\\\sqrt{15}&2 \end{bmatrix} \\ + A &= \begin{bmatrix} 2&-2\\-3&1 \end{bmatrix} \\ + A &= \begin{bmatrix} 2&-4\\4&-6 \end{bmatrix} \\ + \end{align*} + $$ + + \solution + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + \begin{itemize} + \item Compute an eigenvector for the eigenvalue $x = 3$ for the below $(3,3)$-matrix. Note: nobody asks you to compute its characteristic polynomial or to get all of its eigenvalues (Prof did it) + \item Validate that the found eigenvector $v$ is indeed the correct one, that is, that $Av = 3v$ holds. + \end{itemize} + + $$ + A = \begin{bmatrix} 1&0&1\\0&1&-2\\2&4&4 \end{bmatrix} + $$ + + \solution + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + $$ + A = \begin{bmatrix} 1&1\\0&1 \end{bmatrix} + $$ + \begin{itemize} + \item Show that this matrix has only the eigenvalue $1$, twice. + \item Prove that there cannot exist any matrix P such that $P^{-1}DP = A$. Hint: You know how $D$ in $P^{-1}DP$ must look like + \item Find an eigenvector + \end{itemize} + Bonus knowldge: Shear matrices have an eigenspace of dimensionality $d-1$. In the above case the set of all eigenvectors must be $cv, c \in \mathbb{R}$ for some vector $v$.\\ + \solution + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + Show that $$\begin{bmatrix} 2&-4\\\frac{13}{4}&-4 \end{bmatrix}$$ + \begin{itemize} + \item has no real eigenvalue + \item Bonus: what are its complex-valued eigenvalues? + \end{itemize} + + \solution + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + Bonus Matrix: + \begin{itemize} + \item Get its eigenvalues and eigenvectors + \end{itemize} + Note: This is a symmetric one, so you can epect 2 eigenvalues and orthogonal eigenspace + $$ + A = \begin{bmatrix}-2&\sqrt{24}\\\sqrt{24}&8 \end{bmatrix} + $$ + + \solution + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + \begin{itemize} + \item Use numpy to get its eigenvalues and eigenvectors + \item Solve $Ax = (3,17,1/3)$ using numpy + \item Not in exam: + \end{itemize} + Compute the characteristic polynomial for + $$ + A = \begin{bmatrix}1&2&4\\2&-2&1\\3&1&3 \end{bmatrix} + $$ + + \solution + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + What is the cosine of the angle between + $$(6,-6,-4,\sqrt{12}),(6,4,2,\sqrt{25})$$ + + \solution + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + Another 3x3 affine system + \begin{itemize} + \item Show the intermediate result when the first column is the one hot vector $\begin{bmatrix}1\\0\\0\end{bmatrix}$ for the first time + \item Show the intermediate result when the matrix has row echelon form for the first time + \item Get the solution + \end{itemize} + $$ + \begin{align*} + 2x-3y+2z &=-4\\ + 7x+4.5y-1z &=16\\ + 4x +3y+z &=2 + \end{align*} + $$ + + \solution + +\end{homeworkProblem} + + +\end{document} diff --git a/Tutorial 11/INF1004_W11_Tutorial_2200624_Solution.synctex.gz b/Tutorial 11/INF1004_W11_Tutorial_2200624_Solution.synctex.gz new file mode 100644 index 0000000000000000000000000000000000000000..df66b6f46b75ec913f596cf153fdd000590fb09e GIT binary patch literal 69167 zcmaHRbx<5%&@GF*yA#~qoedD&-JReX+zD=hAi)X2-QC^Y-66Oy{x-kwt9tLRw^ds^ zGdpuf?~kuwq>+Aj$AgO}bV0PJyiHfgT6r8hYh zBDSPOZ4x%WOs!rM(*4#DlOhPS@pFIPMa6Gi%fZK%Crby*i=B(>seqTa^8mlct@oRN zE}y5H_qofebF`0>$EmH(&c><$@8`Sg4W*B_wabs&jhCGFr}McYC9+kchpLaGv7(Rn zi}R(N_qmVju8#pitoNHeqxaj4Dy2)2&Zp=1^VF{9oOPp*8>242x4WD7iSV3Nw1Aq2 zqApWR`D@mX_=b3#-xfP>x7@&&S8j;aFSFX1mxV+2-5*c{r=BL;%Z9>f^_07I(tMTsjMJ;mi5N zrQ^rL-KpHiPL)yTyS)O*RFVJV&{8Td*;czyz^~b&zB@+|{~$|$n*gP+x6EI6ma5cN zUF24WJi3%*PMKu_o^G1xOSqfts74geVB4xz+1GdAZQP&;W;r>wR z{nK?*j=xWk+ok`C4dv_ZXih=T86wqD=>UpwZFYP*m87374Pe$NgIy zD`;(OO@aAA9mN~8G4|4=4+}Qlni83)18m8LS_fyX|0~29YhAza_x04~`S+@>&X8cT!_Bw&N23UO!A0%OBS+DW_6+X{lCq1vv5YAt%+cGkhay=&AQ$ev*h!UuFm!V@atp@q}&ZPbqU9<@1)Y~E(?6a z$Q2azd%th0AZ#(iitKnia5UBxeYglQI`YockQ7H$kqXcx;MGO(8>K!kIjH=zwHW|< zzBt&ZlHbivyij-j%hFX;{K9-sc`DFV)C||F`2x*h735Re&U!UgwAJY^)upg@R?dI; 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+\headsep=0.25in + +\linespread{1.1} + +\pagestyle{fancy} +\lhead{\hmwkAuthorName} +\chead{\hmwkClass\ : \hmwkTitle} +\rhead{\firstxmark} +\lfoot{\lastxmark} +\cfoot{\thepage} + +\renewcommand\headrulewidth{0.4pt} +\renewcommand\footrulewidth{0.4pt} + +\setlength\parindent{0pt} + +% +% Create Problem Sections +% + +\newcommand{\enterProblemHeader}[1]{ + \nobreak\extramarks{}{Question \arabic{#1} continued on next page\ldots}\nobreak{} + \nobreak\extramarks{Question \arabic{#1} (continued)}{Question \arabic{#1} continued on next page\ldots}\nobreak{} +} + +\newcommand{\exitProblemHeader}[1]{ + \nobreak\extramarks{Question \arabic{#1} (continued)}{Question \arabic{#1} continued on next page\ldots}\nobreak{} + \stepcounter{#1} + \nobreak\extramarks{Question \arabic{#1}}{}\nobreak{} +} + +\setcounter{secnumdepth}{0} +\newcounter{partCounter} +\newcounter{homeworkProblemCounter} +\setcounter{homeworkProblemCounter}{1} +\nobreak\extramarks{Question \arabic{homeworkProblemCounter}}{}\nobreak{} + +\newenvironment{homeworkProblem}[1][-1]{ + \ifnum#1>0 + \setcounter{homeworkProblemCounter}{#1} + \fi + \section{Question \arabic{homeworkProblemCounter}} + \setcounter{partCounter}{1} + \enterProblemHeader{homeworkProblemCounter} +}{ + \exitProblemHeader{homeworkProblemCounter} +} + +\newcommand{\hmwkClass}{INF 1004 Mathematics 2} +\newcommand{\hmwkTitle}{Tutorial\ \#11} +\newcommand{\hmwkAuthorName}{\textbf{Woon Jun Wei}} +\newcommand{\hmwkStudentID}{\textbf{2200624}} + +\title{ + \vspace{2in} + \textmd{\textbf{\hmwkClass \\ \hmwkTitle}}\\ + \vspace{3in} +} + +\author{\hmwkAuthorName \\ \hmwkStudentID} +\date{\today} + +\renewcommand{\part}[1]{\textbf{\large Part \arabic{partCounter}}\stepcounter{partCounter}\\} + +% +% Various Helper Commands +% + +% Useful for algorithms +\newcommand{\alg}[1]{\textsc{\bfseries \footnotesize #1}} + +% For derivatives +\newcommand{\deriv}[1]{\frac{\mathrm{d}}{\mathrm{d}x} (#1)} + +% For partial derivatives +\newcommand{\pderiv}[2]{\frac{\partial}{\partial #1} (#2)} + +% Integral dx +\newcommand{\dx}{\mathrm{d}x} + +% Alias for the Solution section header +\newcommand{\solution}{\textbf{\large My Solution\\}} +\newcommand{\answer}{\textbf{\large Sample Solutions\\}} + +% Probability commands: Expectation, Variance, Covariance, Bias +\newcommand{\E}{\mathrm{E}} +\newcommand{\Var}{\mathrm{Var}} +\newcommand{\Cov}{\mathrm{Cov}} +\newcommand{\Bias}{\mathrm{Bias}} + +\def\therefore{\boldsymbol{\text{ } +\leavevmode +\lower0.4ex\hbox{$\cdot$} +\kern-.5em\raise0.7ex\hbox{$\cdot$} +\kern-0.55em\lower0.4ex\hbox{$\cdot$} +\thinspace\text{ }}} + +\begin{document} + +\maketitle + +\pagebreak + +\begin{homeworkProblem} + \begin{itemize} + \item Compute the eigenvalues and eigenvectors for the matrices below. + \item For one of these matrices compute the matrix $P$ such that $P^{-1}DP = A$, and verify that $PAP^{-1}$ is the diagonal matrix of the eigenvalues + \end{itemize} + + $$ + \begin{align*} + A &= \begin{bmatrix} 4&\sqrt{15}\\\sqrt{15}&2 \end{bmatrix} \\ + A &= \begin{bmatrix} 2&-2\\-3&1 \end{bmatrix} \\ + A &= \begin{bmatrix} 2&-4\\4&-6 \end{bmatrix} \\ + \end{align*} + $$ + + \solution + + \pagebreak + + \part + + $$ + A = \begin{bmatrix} 4&\sqrt{15}\\\sqrt{15}&2 \end{bmatrix} + $$ + + $$ + \det{\begin{bmatrix} 4-\lambda&\sqrt{15}\\\sqrt{15}&2-\lambda \end{bmatrix}} = 0 + $$ + + $$ + \begin{align*} + f(\lambda) &= \lambda^2 - 6\lambda -7 = 0\\ + &(\lambda - 7)(\lambda + 1) = 0\\ + \lambda &= 7, -1 + \end{align*} + $$ + + \begin{multicols}{2} + $$ + \begin{align*} + \lambda_1 &= -1,\\ + A-(-1)I &= \begin{bmatrix} 5&\sqrt{15}\\\sqrt{15}&3 \end{bmatrix} \\ + &= \begin{bmatrix} \sqrt{15} & 3\\0&0\end{bmatrix} \\ + \sqrt{15}x_0+3x_1 &= 0\\ + x_0 &= -\frac{3}{\sqrt{15}}x_1\\ + \begin{bmatrix}x_0\\x_1\end{bmatrix} &= x_1\begin{bmatrix}-\frac{3}{\sqrt{15}}\\1\end{bmatrix}\\ + \therefore v_1 &= \begin{bmatrix}-3\\\sqrt{15}\end{bmatrix} + \end{align*} + $$ + + \columnbreak + + $$ + \begin{align*} + \lambda_1 &= 7,\\ + A-(7)I &= \begin{bmatrix} -3&\sqrt{15}\\\sqrt{15}&-5 \end{bmatrix} \\ + &= \begin{bmatrix} \sqrt{15} & -5\\0&0\end{bmatrix} \\ + \sqrt{15}x_0-5x_1 &= 0\\ + x_0 &= \frac{5}{\sqrt{15}}x_1\\ + \begin{bmatrix}x_0\\x_1\end{bmatrix} &= x_1\begin{bmatrix}\frac{5}{\sqrt{15}}\\1\end{bmatrix}\\ + \therefore v_1 &= \begin{bmatrix}5\\\sqrt{15}\end{bmatrix} + \end{align*} + $$ + + \end{multicols} + + Case: Symmetric Matix, 2 different eigenvalues, 2 eigenvectors\\ + Eigenspaces are orthogonal, Check: $v_0\cdotv_1 = -15 + \sqrt{15}\sqrt{15} = 0$\\ + + + \textbf{Eigendecomposition}\\ + + $\lambda_1=-1, v_1 = \begin{bmatrix}-3\\\sqrt{15}\end{bmatrix}, \qquad \lambda_2=7, v_2 = \begin{bmatrix}5\\\sqrt{15}\end{bmatrix}$\\ + + $p^{-1} = \begin{bmatrix}v_1&v_2\end{bmatrix} = \begin{bmatrix}-3&5\\\sqrt{15}&\sqrt{15}\end{bmatrix}$\\ + $D = \begin{bmatrix}-1&0\\0&7\end{bmatrix}$\\ + $p = (p^{-1})^{-1} = \frac{1}{-3\sqrt{15}-5\sqrt{15}} \begin{bmatrix}\sqrt{15}&-5\\-\sqrt{15}&-3\end{bmatrix} = \begin{bmatrix}-\frac{1}{8}&\frac{5}{8\sqrt{15}}\\\frac{1}{8}&\frac{3}{8\sqrt{15}}\end{bmatrix}$\\ + Verify: + $$ + \begin{align*} + P^{-1}DP = \begin{bmatrix}-3&5\\\sqrt{15}&\sqrt{15}\end{bmatrix}\begin{bmatrix}-1&0\\0&7\end{bmatrix}\begin{bmatrix}-\frac{1}{8}&\frac{5}{8\sqrt{15}}\\\frac{1}{8}&\frac{3}{8\sqrt{15}}\end{bmatrix} = \begin{bmatrix}4&\sqrt{15}\\\sqrt{15}&2\end{bmatrix} = A\\ + PAP^{-1} = \begin{bmatrix}-\frac{1}{8}&\frac{5}{8\sqrt{15}}\\\frac{1}{8}&\frac{3}{8\sqrt{15}}\end{bmatrix}\begin{bmatrix}4&\sqrt{15}\\\sqrt{15}&2\end{bmatrix}\begin{bmatrix}-\frac{1}{8}&\frac{5}{8\sqrt{15}}\\\frac{1}{8}&\frac{3}{8\sqrt{15}}\end{bmatrix} = \begin{bmatrix}-1&0\\0&7\end{bmatrix} = D + \end{align*} + $$ + + \pagebreak + + \part + + $$ + A = \begin{bmatrix} 2&-2\\-3&1 \end{bmatrix} + $$ + + $$ + \det{\begin{bmatrix} 2-\lambda&-2\\-3&1-\lambda \end{bmatrix}} = 0 + $$ + + $$ + \begin{align*} + f(\lambda) &= \lambda^2 - 3\lambda - 4 = 0\\ + &(\lambda - 4)(\lambda + 1) = 0\\ + \therefore \lambda &= 4, -1 + \end{align*} + $$ + + \begin{multicols}{2} + + $$ + \begin{align*} + \lambda_1 &= -1,\\ + A-(-1)I &= \begin{bmatrix} 3&-2\\-3&2 \end{bmatrix} \\ + &= \begin{bmatrix} 3 & -2\\0&0\end{bmatrix} \\ + 3x_0-2x_1 &= 0\\ + x_0 &= \frac{2}{3}x_1\\ + \begin{bmatrix}x_0\\x_1\end{bmatrix} &= x_1\begin{bmatrix}\frac{2}{3}\\1\end{bmatrix}\\ + \therefore v_1 &= \begin{bmatrix}2\\3\end{bmatrix} + \end{align*} + $$ + + \columnbreak + + $$ + \begin{align*} + \lambda_2 &= 4,\\ + A-(4)I &= \begin{bmatrix} -2&-2\\-3&-3 \end{bmatrix} \\ + &= \begin{bmatrix} -2 & -2\\0&0\end{bmatrix} \\ + -2x_0-2x_1 &= 0\\ + x_0 &= -x_1\\ + \begin{bmatrix}x_0\\x_1\end{bmatrix} &= x_1\begin{bmatrix}-1\\1\end{bmatrix}\\ + \therefore v_2 &= \begin{bmatrix}1\\-1\end{bmatrix} + \end{align*} + $$ + \end{multicols} + + Eigenspaces are not orthogonal, but matrix is not symetric\\ + So even though we get 2 eigenvalues and 2 eigenvectors, it is okay that the eigenvectors are not orthogonal to each other.\\ + + \textbf{Eigendecomposition}\\ + + $\lambda_1=-1, v_1 = \begin{bmatrix}2\\3\end{bmatrix}, \qquad \lambda_2=4, v_2 = \begin{bmatrix}1\\-1\end{bmatrix}$\\ + + $p^{-1} = \begin{bmatrix}v_1&v_2\end{bmatrix} = \begin{bmatrix}2&1\\3&-1\end{bmatrix}$\\ + + $D = \begin{bmatrix}-1&0\\0&4\end{bmatrix}$\\ + + $p = (p^{-1})^{-1} = \frac{1}{-2-3} \begin{bmatrix}-1&-1\\-3&2\end{bmatrix} = \begin{bmatrix}\frac{1}{5}&\frac{1}{5}\\\frac{3}{5}&-\frac{2}{5}\end{bmatrix}$\\ + + $$ + \begin{align*} + P^{-1}DP &= \begin{bmatrix}2&1\\3&-1\end{bmatrix}\begin{bmatrix}-1&0\\0&4\end{bmatrix}\begin{bmatrix}\frac{1}{5}&\frac{1}{5}\\\frac{3}{5}&-\frac{2}{5}\end{bmatrix} = \begin{bmatrix}2&-2\\-3&1\end{bmatrix} = A\\ + PAP^{-1} &= \begin{bmatrix}\frac{1}{5}&\frac{1}{5}\\\frac{3}{5}&-\frac{2}{5}\end{bmatrix}\begin{bmatrix}2&-2\\-3&1\end{bmatrix}\begin{bmatrix}2&1\\3&-1\end{bmatrix} = \begin{bmatrix}-1&0\\0&4\end{bmatrix} = D + \end{align*} + $$ + + \pagebreak + + \part + + $$ + A = \begin{bmatrix} 2&-4\\4&-6 \end{bmatrix} + $$ + + $$ + \det{\begin{bmatrix} 2-\lambda&-4\\4&-6-\lambda \end{bmatrix}} = 0 + $$ + + $$ + \begin{align*} + f(\lambda) &= \lambda^2 +4\lambda +4 = 0\\ + &(\lambda +2)(\lambda +2) = 0\\ + \lambda &= -2, -2 + \end{align*} + $$ + + $$ + \begin{align*} + \lambda_1 &= -2,\\ + A-(-2)I &= \begin{bmatrix} 4&-4\\4&-4 \end{bmatrix} \\ + &= \begin{bmatrix} 4 & -4\\0&0\end{bmatrix} \\ + 4x_0-4x_1 &= 0\\ + x_0 &= x_1\\ + \begin{bmatrix}x_0\\x_1\end{bmatrix} &= x_1\begin{bmatrix}1\\1\end{bmatrix}\\ + \therefore v_1 &= \begin{bmatrix}1\\1\end{bmatrix} + \end{align*} + $$ + + Eigenspace just one dim, but matrix is not symmetric\\ + So it is okay that we do not have 2 eigenvalues nor 2 eigenvectors. (We cant compute D since there is only 1 eigenvector)\\ + + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + \begin{itemize} + \item Compute an eigenvector for the eigenvalue $x = 3$ for the below $(3,3)$-matrix. Note: nobody asks you to compute its characteristic polynomial or to get all of its eigenvalues (Prof did it) + \item Validate that the found eigenvector $v$ is indeed the correct one, that is, that $Av = 3v$ holds. + \end{itemize} + + $$ + A = \begin{bmatrix} 1&0&1\\0&1&-1\\2&4&4 \end{bmatrix} + $$ + + \solution + + $$ + \begin{align*} + A-\lambda I &= \begin{bmatrix} 1-\lambda&0&1\\0&1-\lambda&-1\\2&4&4-\lambda \end{bmatrix} \\ + f{\lambda} &= - \lambda^3 + +6\lambda^2 - 11\lambda + 6 = 0\\ + f(3) &= 3^3 - 6\cdot3^2 + 11\cdot3 + 6 = 0\\ + \end{align*} + $$ + When $\lambda = 3&,\\ + + $$ + \begin{align*} + A-3I &= \begin{bmatrix} -2&0&1\\0&-2&-1\\2&4&1 \end{bmatrix} \\ + &= \begin{bmatrix} -2&0&1\\0&-2&-1\\0&0&0 \end{bmatrix} + \end{align*} + $$ + + $1^{st}$ row: $-2x_0 + x_2 = 0 \Rightarrow x_0 = \frac{1}{2}x_2$\\ + $2^{nd}$ row: $-2x_1 - x_2 = 0 \Rightarrow x_1 = -\frac{1}{2}x_2$\\ + + $$ + \begin{align*} + \begin{bmatrix}x_0\\x_1\\x_2\end{bmatrix} &= x_2 \begin{bmatrix}\frac{1}{2}\\-\frac{1}{2}\\1\end{bmatrix}\\ + \therefore v &= \begin{bmatrix}1\\-1\\2\end{bmatrix} + \end{align*} + $$ + + Verify: + $$ + \begin{bmatrix}1&0&1\\0&1&-1\\2&4&4 \end{bmatrix}\begin{bmatrix}1\\-1\\2\end{bmatrix} = 3\begin{bmatrix}1\\-1\\2\end{bmatrix} + $$ + + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + $$ + A = \begin{bmatrix} 1&1\\0&1 \end{bmatrix} + $$ + \begin{itemize} + \item Show that this matrix has only the eigenvalue $1$, twice. + \item Prove that there cannot exist any matrix P such that $P^{-1}DP = A$. Hint: You know how $D$ in $P^{-1}DP$ must look like + \item Find an eigenvector + \end{itemize} + Bonus knowldge: Shear matrices have an eigenspace of dimensionality $d-1$. In the above case the set of all eigenvectors must be $cv, c \in \mathbb{R}$ for some vector $v$.\\ + \solution + + $$ + A = \begin{bmatrix} 1&1\\0&1 \end{bmatrix} + $$ + + $$ + A - \lambda I = \begin{bmatrix} 1-\lambda&1\\0&1-\lambda \end{bmatrix} + $$ + + $$ + f(\lambda) = (\lambda -1)^2 = 0 + $$ + + So only $\lambda = 1$ is a solution. It has 2 eigenvalues equal to 1 (\textit{eigenvalue `1` has algebraic mutiplicity of 2})\\ + + Suppose P exists such that $P^{-1}DP = A$, then $D = \begin{bmatrix}1&0\\0&1\end{bmatrix}$\\ + $P^{-1}DP = P^{-1}IP = P^{-1}P = I$\\ + But A is not the identity matrix. So there cannot exist any P as assumed above (Proof by Contradiction) + + $$ + A-I = \begin{bmatrix} o&1\\0&0 \end{bmatrix} + $$ + + $$0\cdot x_0 + 1 \cdot x_1 = 0 \Rightarrow x_1 = 0$$ + $$\begin{bmatrix}x_0\\x_1\end{bmatrix} = \begin{bmatrix}1\\0\end{bmatrix}$$ + $$\therefore \text{ eigenvector } v = \begin{bmatrix}1\\0\end{bmatrix}$$ + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + Show that $$\begin{bmatrix} 2&-4\\\frac{13}{4}&-4 \end{bmatrix}$$ + \begin{itemize} + \item has no real eigenvalue + \item Bonus: what are its complex-valued eigenvalues? + \end{itemize} + + \solution + + $$ + A-\lambda I = \begin{bmatrix} 2-\lambda&-4\\ \frac{13}{4}&-4-\lambda \end{bmatrix} + $$ + + $$f(\lambda) = (\lambda - 2)(\lambda + 4) = \lambda^2 +2\lambda +5 = 0$$ + $$\lambda = \frac{-(2)\pm \sqrt{(2)^2 - 4\cdot 5}}{2} = \frac{-2\pm 2\sqrt{5}}{2} = \frac{-2\pm\sqrt{-1}\sqrt{16}}{2} = \frac{-2\pm i(4)}{2}=-1\pm2i$$ + + There are no real eigenvalues. + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + Bonus Matrix: + \begin{itemize} + \item Get its eigenvalues and eigenvectors + \end{itemize} + Note: This is a symmetric one, so you can expect 2 eigenvalues and orthogonal eigenspace + $$ + A = \begin{bmatrix}-2&\sqrt{24}\\\sqrt{24}&8 \end{bmatrix} + $$ + + \solution + + $$ + A = \begin{bmatrix}-2&\sqrt{24}\\\sqrt{24}&8 \end{bmatrix} + $$ + + $$ + A-\lambda I = \begin{bmatrix}-2-\lambda&\sqrt{24}\\\sqrt{24}&8-\lambda \end{bmatrix} + $$ + + $$ + f(\lambda) = (\lambda + 2)(\lambda - 8) = \lambda^2 - 6\lambda - 40 = (\lambda - 10)(\lambda + 4) = 0 + $$ + So $\lambda = 10$ or $\lambda = -4$ + + \begin{multicols}{2} + + $$ + \begin{align*} + A - -4I &= \begin{bmatrix}2&\sqrt{24}\\\sqrt{24}&12 \end{bmatrix}\\ + &= \begin{bmatrix}\sqrt{24}&12\\0&0\end{bmatrix}\\ + \sqrt{24}x_0+12x_1 &= 0\\ + x_0 &= -\frac{12}{\sqrt{24}}x_1\\ + \begin{bmatrix}x_0\\x_1\end{bmatrix} &= \begin{bmatrix}-\frac{12}{\sqrt{24}}\\1\end{bmatrix}\\ + \therefore v_1 &= \begin{bmatrix}-12\\\sqrt{24}\end{bmatrix} + \end{align*} + $$ + + \columnbreak + + $$ + \begin{align*} + A - 10I &= \begin{bmatrix}-12&\sqrt{24}\\\sqrt{24}&-2 \end{bmatrix}\\ + &= \begin{bmatrix}-12&\sqrt{24}\\0&0\end{bmatrix}\\ + -12x_0+\sqrt{24}x_1 &= 0\\ + x_0 &= \frac{\sqrt{24}}{12}x_1\\ + \begin{bmatrix}x_0\\x_1\end{bmatrix} &= \begin{bmatrix}\frac{\sqrt{24}}{12}\\1\end{bmatrix}\\ + \therefore v_2 &= \begin{bmatrix}\sqrt{24}\\12\end{bmatrix} + \end{align*} + $$ + \end{multicols} + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + \begin{itemize} + \item Use numpy to get its eigenvalues and eigenvectors + \item Solve $Ax = (3,17,1/3)$ using numpy + \item Not in exam: + \end{itemize} + Compute the characteristic polynomial for + $$ + A = \begin{bmatrix}1&2&4\\2&-2&1\\3&1&3 \end{bmatrix} + $$ + + \solution + + $$ + A-\lambda I = \begin{bmatrix}1-\lambda&2&4\\2&-2-\lambda&1\\3&1&3-\lambda \end{bmatrix} + $$ + + $$ + f(\lambda) = -\lambda^3 + 2\lambda^2 +22\lambda+19 + $$ + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + What is the cosine of the angle between + $$(6,-6,-4,\sqrt{12}),(6,4,2,\sqrt{25})$$ + + \solution + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + Another 3x3 affine system + \begin{itemize} + \item Show the intermediate result when the first column is the one hot vector $\begin{bmatrix}1\\0\\0\end{bmatrix}$ for the first time + \item Show the intermediate result when the matrix has row echelon form for the first time + \item Get the solution + \end{itemize} + $$ + \begin{align*} + 2x-3y+2z &=-4\\ + 7x+4.5y-1z &=16\\ + 4x +3y+z &=2 + \end{align*} + $$ + + \solution + +\end{homeworkProblem} + + +\end{document} diff --git a/Tutorial 11/tut5_eigenvalues_etc.pdf b/Tutorial 11/tut5_eigenvalues_etc.pdf new file mode 100644 index 0000000000000000000000000000000000000000..80216a28bcd2eee50b07a9a77c373ce09b3b4943 GIT binary patch literal 109251 zcmbrmWmKHo(lv^^26uOD8h3ZMK#<@Tg1fs15AN=P;FjPHL4yQ$2qCybe`N1{-uK+I 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\setcounter{homeworkProblemCounter}{#1} + \fi + \section{Question \arabic{homeworkProblemCounter}} + \setcounter{partCounter}{1} + \enterProblemHeader{homeworkProblemCounter} +}{ + \exitProblemHeader{homeworkProblemCounter} +} + +\newcommand{\hmwkClass}{INF 1004 Mathematics 2} +\newcommand{\hmwkTitle}{Probability Practice Exercises} +\newcommand{\hmwkAuthorName}{\textbf{Woon Jun Wei}} +\newcommand{\hmwkStudentID}{\textbf{2200624}} + +\title{ + \vspace{2in} + \textmd{\textbf{\hmwkClass \\ \hmwkTitle}}\\ + \vspace{3in} +} + +\author{\hmwkAuthorName \\ \hmwkStudentID} +\date{\today} + +\renewcommand{\part}[1]{\textbf{\large Part \arabic{partCounter}}\stepcounter{partCounter}\\} + +% +% Various Helper Commands +% + +% Useful for algorithms +\newcommand{\alg}[1]{\textsc{\bfseries \footnotesize #1}} + +% For derivatives +\newcommand{\deriv}[1]{\frac{\mathrm{d}}{\mathrm{d}x} (#1)} + +% For partial derivatives +\newcommand{\pderiv}[2]{\frac{\partial}{\partial #1} (#2)} + +% Integral dx +\newcommand{\dx}{\mathrm{d}x} + +% Alias for the Solution section header +\newcommand{\solution}{\textbf{\large My Solution\\}} + +% Probability commands: Expectation, Variance, Covariance, Bias +\newcommand{\E}{\mathrm{E}} +\newcommand{\Var}{\mathrm{Var}} +\newcommand{\Cov}{\mathrm{Cov}} +\newcommand{\Bias}{\mathrm{Bias}} + +\begin{document} + +\maketitle + +\pagebreak + +\begin{homeworkProblem} +A jar contains four coins: a nickel (5 cents), a dime (10 cents), a quarter (25 cents), and a half-dollar (50 cents). +Three coins are randomly selected without replacement from the jar. +\begin{enumerate}[label=(\alph*)] + \item List all the possible outcomes in sample space S. + \item What is the probabilty that the selection will contain the half-dollar? + \item What is the probability that the total amount drawn will equal 65 cents or less? + \item Which interpretations of probability are you applying? +\end{enumerate} + +\solution + +\part + +$$S = \{(5,10,25), (5,25,50), (10, 25,50), (5,10,50)\}$$ + +\part + +$$P(\text{half dollar}) = \frac{3}{4}$$ + +\part + +Sum of the outcomes: +$$S = \{40, 80, 85, 65\}$$ + +$$P(\text{total} \leq 65) = \frac{2}{4} = \frac{1}{2}$$ + +\part + +Classical interpretation. + + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + In a genetics experiment, the researcher mated two Drosophila fruit flies + and observed the traits of 300 offsprint. The results are shown in the table + + \includegraphics[width=\textwidth]{q2.png} + One of these offspring is randomly selected and observed for the two genetic traits + + \begin{enumerate}[label=(\alph*)] + \item What is the probability that the fly has normal eye color and normal wing size? + \item What is the probability that the fly has vermillion eyes? + \item What is the probability that the fly has either vermillion eyes or miniature wings, or both? + \item Which interpretations of probability are you applying? +\end{enumerate} + +\solution + +\part + +$$P(\text{Normal Eye Color} \cap \text{Normal Wing Size}) = \frac{140}{300} = \frac{7}{15} = 0.46667$$ + +\part + +$$P(\text{Vermillion Eyes}) = \frac{3+151}{300} = \frac{77}{150} = 0.513$$ + +\part + +$$ +\begin{align} +P(\text{vermillion eyes } \cup \text{ miniature wings})&= P(\text {vermillion eyes}) + P(\text{miniature wings}) - P(\text{vermillion eyes} \cap \text{miniature wings})\\ + &= \frac{154}{300} + \frac{157}{300} - \frac{151}{300} + &= \frac{160}{300} + &= \frac{8}{15} +\end{align} +$$ + +\part + +Relative frequency interpretation. + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + City residents were surgeyed recently to determine readership of newspapers. + 50\% of the residents read the morning paper, 60\% read the evening paper and + 20\% read both newspapers. Find the probability that a resident selected + at random reads the morning or evening paper + + \solution + + Given: \\ + $P(\text{morning}) = 0.5$\\ + $P(\text{evening}) = 0.6$\\ + $P(\text{both}) = 0.2 = P(\text{morning } \cap \text{ evening})$\\ + + To find: $P(\text{morning or evening})$\\ + + $$ + \begin{align} + P(\text{morning} \cup \text{evening}) &= P(\text{morning}) + P(\text{evening}) - P(\text{morning} \cap \text{evening})$\\ + &= 0.5 + 0.6 - 0.2\\ + &= 0.9 + \end{align} + $$ + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + The experience of a train passenger is that the train is cancelled with probability 0.02 + When it does run, it has probability of 0.9 of being on time. + \begin{enumerate}[label=(\alph*)] + \item Find the probability that the train runs on time + \item Given that the train is not on time, find the probability that it has been cancelled + \end{enumerate} + + \solution + + Given:\\ + P(\text{cancelled}) = 0.02\\ + P(\text{cancelled}^c) = 0.98\\ + P(\text{on time} | \text{cancelled}^c) = 0.9\\ + + \part + + $$P(\text{on time}) = 0.98 \cdot 0.9 = 0.882$$ + + \part + + From part (a): + $$P(\text{on time}^c) = 1 - 0.882 = 0.118$$\\ + + $$ + \begin{align} + P(\text{cancelled} | \text{on time}^c) &= \frac{P(\text{cancelled} \cap \text{on time}^c)}{P(\text{on time}^c)}\\ + &= \frac{0.02\cdot 1}{0.02 \cdot 1 + 0.98 \cdot 0.1}\\ + &= 0.1694915\\ + &\approx 0.1694 + \end{align} + $$ + + Draw the probability tree diagram to visualise.\\ + Note: Once the train is cancelled, there is \textbf{NO WAY} the train can be on time,\\ + Therefore, $P(\text{on time} | \text{cancelled}) = 0$ and $P(\text{on time}^c | \text{cancelled}) = 1$. + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + In a group of students 60\% are female and 40\% are male. + A third of the female students study french but only a quarter of the male students study french + \\ + A student is chosen at random from the group. + \begin{enumerate}[label=(\alph*)] + \item Show that the probability that the chosen student is female and studies french is 0.2 + \item Calculate the probability that the chosen student studies french + \item Given that the chosen student does study french, calculate the conditional probability that the chosen student is female + \end{enumerate} + + \solution + + Let f: female\\ + fr: studies french + \\\\ + Given: + $P(f) = 0.6$\\ + $P(f^c) = 0.4$\\ + $P(fr) = 0.3$\\ + $P(fr^c) = 0.7$\\ + $P(fr | f) = 0.3$\\ + $P(fr^c | f) = 0.7$\\ + $P(fr | f^c) = 0.25$\\ + $P(fr^c | f^c) = 0.75$\\ + + Draw the probability tree to help solve the problem + + \part + + $$ + P(f \cap fr) = P(f) \cdot P(fr | f) = 0.6 \cdot \frac{1}{3} = 0.2 + $$ + + \part + + $$P(fr) = 0.2 + (0.4 \cdot 0.25) = 0.2 + 0.1 = 0.3$$ + + \part + + $$P(f | fr) = \frac{P(f \cap fr)}{P(fr)} = \frac{0.2}{0.3}$$ + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} +Of the patients in an emergency unit, 30\% have a sports injury. Pass records +for this emergency unit suggest that :\\ + +A patient with sports injury has a probability of 0.2 of being admitted to hospital\\ +\\ +A patient who doesn’t have a sports injury has a probability of 0.4 of being admitted to +hospital\\ +\\ +Now, a patient is chosen at random from those in the emergency unit.\\ + + \begin{enumerate}[label=(\alph*)] + \item What is the probability that the chosen patient has a sports injury + \item What is the probability that the chosen patient has a sports injury and is admitted to hospital + \item Show that the probability that the chosen patient is admitted to hospital is 0.34 + \item Given that the chosen patient is admitted to hospital, find the conditional probability that the patient has a sports injury + \end{enumerate} + + \solution + + Given:\\ + $P(\text{sports injury}) = 0.3$\\ + $P(\text{sports injury}^c) = 0.7$\\ + $P(\text{admitted} | \text{sports injury}) = 0.2$\\ + $P(\text{admitted} | \text{sports injury}^c) = 0.4$\\ + + \part + + $$P(\text{sports injury}) = 0.3$$ + + \part + + $$P(\text{sports injury} \cap \text{admitted}) = P(\text{sports injury}) \cdot P(\text{admitted} | \text{sports injury}) = 0.3 \cdot 0.2 = 0.06$$ + + \part + + $$P(\text{admitted}) = P(\text{sports injury} \cap \text{admitted}) + P(\text{sports injury}^c \cap \text{admitted}) = 0.3 \cdot 0.2 + 0.7 \cdot 0.4 = 0.34$$ + + \part + + $$P(\text{sports injury} | \text{admitted}) = \frac{P(\text{sports injury} \cap \text{admitted})}{P(\text{admitted})} = \frac{0.06}{0.34} = 0.17647 \approx 0.18$$ + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + +Darren and Charles are friends who often go to the cinema together. On such visits there +is a probability of 0.4 that Darren will buy popcorn. +The probability that Charles will buy +popcorn is 0.7 if Darren buys popcorn and 0.35 if he doesn’t. + +When Darren and Charles visit the cinema together: + \begin{enumerate}[label=(\alph*)] + \item Find the probability that both buy popcorn + \item Show that the probability that neither buys popcorn is 0.39 + \item Find the probability that exactly one of them buys popcorn + \end{enumerate} +Sarah sometimes join Darren and Charles on their cinema visits. On these occasions, +the probability that sarah buys popcorn is 0.55 if both +of charles and darren buy popcorn and 0.25 if exactly one of darren and charles buys popcorn + +When Darren, Sarah and Charles visit the cinema together: + + \begin{enumerate}[label=(\alph*)] + \item What is the probability that the three of them buy popcorn + \item Charles and sarah buy popcorn but Daren Doesn't + \end{enumerate} + + \solution + + Given:\\ + $P(\text{Darren buys popcorn}) = 0.4$\\ + $P(\text{Charles buys popcorn} | \text{Darren buys popcorn}) = 0.7$\\ + $P(\text{Charles buys popcorn} | \text{Darren doesn't buy popcorn}) = 0.35$\\ + $P(\text{Darren doesn't buy popcorn}) = 0.6$\\ + + \part + + $$ + \begin{align} + & P(\text{Darren buys popcorn} \cap \text{Charles buys popcorn}) \\ + &= P(\text{Darren buys popcorn}) \cdot P(\text{Charles buys popcorn} | \text{Darren buys popcorn})\\ + &= 0.4 \cdot 0.7\\ + &= 0.28 + \end{align} + $$ + + \part + + $$ + \begin{align} + & P(\text{Darren doesn't buy popcorn} \cap \text{Charles doesn't buy popcorn}) \\ + &= P(\text{Darren doesn't buy popcorn}) \cdot P(\text{Charles doesn't buy popcorn} | \text{Darren doesn't buy popcorn})\\ + &= 0.6 \cdot 0.65\\ + &= 0.39 + \end{align} + $$ + + + \part + + $$ + \begin{align} + & P(\text{Darren buys popcorn} \cap \text{Charles doesn't buy popcorn} \text{ or } \text{Charles buys popcorn} \cap \text{Darren doesn't buy popcorn}) \\ + &= P(\text{Darren buys popcorn}) \cdot P(\text{Charles doesn't buy popcorn} | \text{Darren buys popcorn}) \\ + &+ P(\text{Charles buys popcorn}) \cdot P(\text{Darren doesn't buy popcorn} | \text{Charles buys popcorn})\\ + &= 0.4 \cdot 0.3 + 0.6 \cdot 0.35\\ + &= 0.33 + \end{align} + $$ + + \part + + Given: + $P(\text{Sarah buys popcorn} | \text{Darren buys popcorn} \cap \text{Charles buys popcorn}) = 0.55$\\ + $P(\text{Sarah buys popcorn} | \text{Darren buys popcorn} \cap \text{Charles doesn't buy popcorn}) = 0.25$\\ + $P(\text{Sarah buys popcorn} | \text{Darren doesn't buy popcorn} \cap \text{Charles buys popcorn}) = 0.25$\\ + + $$ + \begin{align} + P(\text{3 buy popcorn}) &= P(\text{Darren buys popcorn} \cap \text{Charles buys popcorn} \cap \text{Sarah buys popcorn}) \\ + &= 0.4 \cdot 0.7 \cdot 0.55\\ + &= 0.154 + \end{align} + $$ + + \part + + $$ + \begin{align} + P(\text{Charles buys popcorn} \cap \text{Sarah buys popcorn} \cap \text{Darren doesn't buy popcorn}) &= 0.6 \cdot 0.35 \cdot 0.25\\ + &= 0.0525 + \end{align} + $$ + + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + + +Following a flood, 120 tins were recovered from Lee’s corner shop. +Unfortunately, the water had washed off all the labels. +Of the tins, 50 contained pet food, 20 contained peas, 35 +contained beans and the rest contained soup. + + \begin{enumerate}[label=(\alph*)] +\item Lee selects a tin at random, what is the probability that it contains soup? +\item Lee selects a tin at random, what is the probability that it doesn’t contain pet food +\item Lee selects two tins at random (without replacement), what is the probability that both + contains peas +\item Lee selects two tins at random (without replacement), what is the probability that one +contains pet food and the other contains peas +\item Lee selects 3 tins at random (without replacement). What is the probability that one +contains pet food, one contains peas and one contains beans. +\item Find the probability that Lee will have to open more than two tins before he finds one +that doesn’t contain pet food. + \end{enumerate} + + \solution + + \part + + Soup left: 120 - 50 - 20 - 35 = 15\\ + + $$P(\text{soup}) = \frac{15}{120} = 0.125$$ + + \part + + $$P(\text{pet food}^c) = \frac{120-50}{120} = 0.58333$$ + + \part + + $$P(\text{peas} \cap \text{peas}) = \frac{20}{120} \cdot \frac{19}{119} = 0.00254$$ + + \part + + $$P(\text{pet food} \cap \text{peas}) = \frac{50}{120} \cdot \frac{20}{119} \cdot 2 = 0.140056$$ + + \part + + $$P(\text{pet food} \cap \text{peas} \cap \text{beans}) = \frac{50}{120} \cdot \frac{20}{119} \cdot \frac{35}{118} \cdot 3! = 0.12462612164 \approx 0.125$$ + + \part + + What are the chances that it's just + + $$\frac{50}{120} \cdot \frac{49}{119} = 0.1715686 \approx 0.172$$ + + + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + A school employs 75 teachers. The following table summarizs their length of service at the school, classified by gender. + + \begin{tabular}{|c|c|c|c|} + \hline + & $<$ 3 years & 3 years to 8 years & $>$ 8 years\\ + \hline + Female & 12 & 20 & 13\\ + \hline + Male & 8 & 15 & 7\\ + \hline + \end{tabular} +\\ + \begin{enumerate}[label=(\alph*)] + \item Find the probability that a randomly selected teacher is a female + \item Find the probability that a randomly selected teacher is female given that the teacher has more than 8 years service + \item Find the probability that a randomly selected teacher is female given that the teacher has less than 3 years service + \item State, giving a reason, whether or not the event of selecting a female teacher is independent of the event of selecting a teacher with less than 3 years service + \end{enumerate} + + \solution + + \part + + $$P(\text{female}) = \frac{12+20+13}{75} = \frac{45}{75} = 0.6$$ + + \part + + $$P(\text{female} | >\text{8 years}) = \frac{P(\text{female} \cap >\text{8 years})}{P(>\text{8 years})} = \frac{\frac{13}{75}}{\frac{20}{75}} = 0.65$$ + + \part + + $$P(\text{female}|< \text{3 years}) = \frac{P(\text{female} \cap < \text{3 years})}{P(< \text{3 years})} = \frac{\frac{12}{75}}{\frac{20}{75}} = 0.6$$ + + \part + + Test for independence: $$P(\text{female}| < \text{3 years}) = P(\text{female})$$\\ + Since both are 0.6, the events are independent. + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + +A group of students bought, in total, 25 items of clothing at two shops: Mango and +Iora. The following table shows how many tops, jeans and sweaters were +bought at each of the two shops:\\ + \begin{tabular}{|c|c|c|c|} + \hline + & Top & Jeans & Sweaters\\ + \hline + Mango & 3 & 7 & 5\\ + \hline + Iora & 2 & 5 & 3\\ + \hline + \end{tabular} + \\ + One item of clothing is chosen at random from these 25 items:\\ + \begin{enumerate}[label=(\alph*)] + \item What is the probability that the chosen item is a top? + \item What is the probability that the chosen item was bought from Iora? + \item What is the probability that the chosen item is a top and was bought from Iora? + \item State with a reason whether the events (chosen item is a top) and (chosen item was bought from iora) are independent. + \item Given that the chosen item is not a top, find the conditional probability that it was + \end{enumerate} + \solution + + \begin{tabular}{|c|c|c|c|c|} + \hline + & Top & Jeans & Sweaters & Total\\ + \hline + Mango & 3 & 7 & 5 & 15\\ + \hline + Iora & 2 & 5 & 3 & 10\\ + \hline + Total & 5 & 12 & 8 & 25\\ + \hline + \end{tabular} + \\\\ + \part + + $$P(\text{top}) = \frac{5}{25} = 0.2$$ + + \part + + $$P(\text{Iora}) = \frac{10}{25} = 0.4$$ + + \part + + $$P(\text{top} \cap \text{Iora}) = \frac{2}{25} = 0.08$$ + + \part + + $$P(\text{top} | \text{Iora}) = \frac{P(\text{top} \cap \text{Iora})}{P(\text{Iora})} = \frac{0.08}{0.4} = 0.2$$ + + Test for independence: $$P(\text{top}) = P(\text{top} | \text{Iora})$$\\ + Since both are 0.2, the events are independent. + \\\\ + \part + + $$P(\text{Iora} | \text{top}^c) = \frac{P(\text{Iora} \cap \text{top}^c)}{P(\text{top}^c)} = \frac{0.4 \cdot 0.8}{0.8} = 0.4$$ + + + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} +A volunteer for the Health Literacy Center was investigating the attitudes of students towards +smoking on campus. A random sample of 730 students from all four grade levels was taken. +Each student was given the statement "Smoking is dangerous for your health" and asked +whether they agreed, had no opinion or disagreed. The following contingency table summarizes the results. + + \begin{enumerate}[label=(\alph*)] + \item Complete the following table knowing that + \begin{enumerate} + \item 387 Male students participated + \item 277 students agreed and 97 students had no opinion + \item 154 from those who agreed were female + \item Only 36 from the males who participated had no opinion + \end{enumerate} + \item Among the participants, what is the probability that a student agreed P (A)? + \item Show whether the events of students agreeing(A) and being a female student (F) are independent + \item What is the probability that a student is a female given that the student had no opinion $P(F|N)$? What is the probability that a student disagreed, given that the student is Male $P(D|M)$? + \item Use Bayes' Theorem to calculate the probability $P(M|A)$ + \end{enumerate} + + \solution\\ + + \part + + \begin{tabular}{|c|c|c|c|c|} + \hline + & Agree (A) & No Opinion (N) & Disagree (D) & Totals\\ + \hline + Male (M) & 123 & 36 & 228 & 387 \\ + \hline + Female (F) & 154 &61 &128 &343 \\ + \hline + Totals &277 & 97 & 356 &730 \\ + \hline + \end{tabular} + \\\\ + \part + + $$P(A) = \frac{277}{730} = 0.38$$ + + \part + + $$P(A|F) = \frac{P(A \cap F)}{P(F)} = \frac{0.21}{0.47} = 0.45$$\\ + Test for independence: $$P(A) = P(A|F)$$\\ + Since $P(A) = 0.38$ and $P(A|F) = 0.45$, the events are not independent. + + \part + + $$P(F|N) = \frac{P(F \cap N)}{P(N)} = \frac{\frac{61}{730}}{\frac{97}{730}} = 0.63$$ + + $$P(D|M) = \frac{P(D \cap M)}{P(M)} = \frac{\frac{228}{730}}{\frac{387}{730}} = 0.59$$ + + \part + + $$P(M|A) = \frac{P(M \cap A)}{P(A)} = \frac{\frac{123}{730}}{\frac{277}{730}} = 0.44$$ + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} +Letter A is sent by first-class post and has a probability of 0.9 of being delivered next day. Letter B is sent by second-class post and has a probability of only 0.3 if being delivered the next day. + \begin{enumerate}[label=(\alph*)] + \item Find the probability that both letters are delivered the next day + \item Find the probability that neither letter is delivered the next day + \item Find the probability that at least one of the letters is delivered the next day + \item Given that at least one of the letters is delivered the next day, find the probability that letter A is delivered the next day + \end{enumerate} + + \solution\\ + + Draw the Probability tree diagram to solve the problem. + + \part + + $$P(A \cap B) = P(A) \cdot P(B) = 0.9 \cdot 0.3 = 0.27$$ + + \part + + $$P(A^c \cap B^c) = P(A^c) \cdot P(B^c) = 0.1 \cdot 0.7 = 0.07$$ + + \part + + $$P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.9 + 0.3 - 0.27 = 0.93$$ + OR, Using a tree diagram:\\ + $$P(A \cup B) = 0.9 \cdot 0.3 + 0.9 \cdot 0.7 + 0.1 \cdot 0.3 = 0.93$$ + + \part + + $$P(a|\text{at least 1 letter}) = \frac{P(a \cap \text{at least 1 letter})}{P(\text{at least 1 letter})} = \frac{0.9 \cdot 0.3 + 0.9 \cdot 0.7}{0.93} = 0.9677419 \approx 0.9677$$ + +\end{homeworkProblem} +\pagebreak + +\begin{homeworkProblem} + \textbf{ANSWERS} + \begin{enumerate} + \item (b) $\frac{3}{4}$, $c = \frac{1}{2}$, classical interpretation + \item $\frac{7}{15}$, $\frac{77}{150}$, $\frac{8}{15}$, Relative frequency interpretation + \item 0.9 + \item 0.882, 0.1694 + \item 0.2,0.3, 0.2/0.3 + \item 0.3, 0.06, 0.34, 0.17647 + \item 0.28, 0.39, 0.33,0.154, 0.0525 + \item 0.125, 0.0583, 0.0266, 0.140, 0.125, 0.172 + \item 0.6, 0.65, 0.6, independent + \item $\frac{1}{5}$, $\frac{2}{5}$, $\frac{2}{25}$, independent, $\frac{2}{5}$ + \item B)P(A) = $\frac{377}{730}$, C) not Independent, D) $\frac{61}{97}, \frac{228}{387}$, E) 0.44404 + \item A) 0.27, B)0.07, C)0.93, D)0.9677 + \end{enumerate} + + 11a) + + \begin{tabular}{|c|c|c|c|c|} + \hline + & Agree (A) & No Opinion (N) & Disagree (D) & Totals\\ + \hline + Male (M) & 123 & 36 & 228 & 387 \\ + \hline + Female (F) & 154 &61 &128 &343 \\ + \hline + Totals &277 & 97 & 356 &730 \\ + \hline + \end{tabular} +\end{homeworkProblem} + +\end{document} diff --git a/Exercises/Probability/Probability_Practice_Blank.pdf b/Exercises/Probability/Probability_Practice_Blank.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b035380e793b6cc4b9543778b67b691e309f3452 GIT binary patch literal 121147 zcma%iWmH^UlWybCSmTyJLvVrz2n1=|p^?U26Fj)POK^7y(zpjFxJz&d7A(OfxLw}w zn|o(w&HT9a>#SP)Y^hyO)l;X|p@T|Dv2wEU0O_V?`xk+{U=Faoi4{;#5Xi1zZfEKI z5zNKI5B|3R*`+?&IGa0y*`;iZoy{f8P3_Ihfx^N-Cuc`;g>|KyR7 zU+KF=;#k)evrT_sU05L68Ha8;u`{ME$={U^t4sw9${yT_*Q1%BcEs3bhYGW zO2@tLD0@GEzD-^m??F*#5FESY(#&uuk6-Xlt&;nbKPAupTNN@1DKPtq+2Yv;0x zwout(;>@_9XS08F7=9CMB1mYXLLqAq;pc49r1IdUj(4A~UqRlncFwn%6s5VxQMuI5 z#H%Tb9$~;Kg1I^;1~uO4t9;)(#KsmRlpl{h==bWL_Z~Gk)5H&GZfEu%)_8gP$C8}y z{-;U#IJy46P5M=R*lvXb)N#P#UK9|yNh7DhE{aJj8B{C-LgFvtpTd7#|Mjgjrs}hg 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