\documentclass[12pt]{article} \usepackage{fullpage,amsmath,amssymb,graphicx,enumitem,color} \def\full{n} \def\fullcut{\if\full y \newpage\null \fi} %\topmargin -1.3in %\textheight 9.9in \newcommand{\bbQ}{{\mathbb{Q}}} \newcommand{\bbR}{{\mathbb R}} \begin{document} \noindent Name: $\underline{\hspace{2.5in}}$ \hfill Student ID: $\underline{\hspace{1.5in}}$ \vfil \begin{center} {\Large UNIVERSITY OF TORONTO}\\ \vskip 2mm {\large Faculty of Arts and Sciences}\\ \vskip 2mm {\Large DECEMBER EXAMINATIONS 2014}\\ \vskip 2mm {\Large Math 240H1 Algebra I --- Makeup Final Exam} \vskip 2mm Dror Bar-Natan\par February 2015\par \end{center} \vfil {\bf Solve 4 of the following 5 questions.} If you solve more than 4 questions indicate very clearly below which are the ones that you want marked, or else a random one will be left out. Please write your answers within this booklet. You may also use the back side of the pages for answers and you may continue answers in the blank pages provided at the end of this notebook (though indicate ``continued on page $\ldots$'' when you do so). You may mark pages ``draft'' and use them for scratch work. The questions carry equal weight though different parts of the same question may be weighted differently. There are \pageref{goodluck} pages in this exam booklet. \vfil \noindent{\bf Duration. } You have 3 hours to write this exam. \vskip 2mm \noindent{\bf Allowed Material. } Stationary and basic calculators, not capable of displaying text or sounding speech. A stuffed animal for personal comfort. \vfil \centerline{{\bf Good Luck!}} \vfil \begin{center} For grading use --- circle the 4 questions that you want marked:\par\bigskip \renewcommand{\arraystretch}{3} %\newcommand{\mybox}{\vbox to 0.1in{\vfill\hbox to 1.5in{\hfil}}} \newcommand{\mybox}{\hbox to 1.0in{\hfil}} \begin{tabular}{|c|r|c|r|} \hline 1 & \mybox /25 & 4 & \mybox /25 \\ \hline 2 & \mybox /25 & 5 & \mybox /25 \\ \hline 3 & \mybox /25 & & \\ \hline\hline Total & \multicolumn{3}{|r|}{\mybox /100} \\ \hline \end{tabular} \end{center} \newpage \noindent{\bf Problem 1.} Let $A$ be a matrix in $M_{2\times 2}(F)$. Let $T\colon M_{2\times 2}(F)\to M_{2\times 2}(F)$ be defined by $T(B)=A^TBA$. \begin{enumerate} \item Show that $T$ is a linear transformation. \item Find the matrix representing $T$ relative to the basis \[ \left\{\begin{pmatrix}1&0\\0&0\end{pmatrix}, \begin{pmatrix}0&1\\0&0\end{pmatrix}, \begin{pmatrix}0&0\\1&0\end{pmatrix}, \begin{pmatrix}0&0\\0&1\end{pmatrix}\right\}, \] which is taken as a basis of both the domain space of $T$ and the target space of $T$. \end{enumerate} \vskip 1mm \par\noindent{\small {\bf Tip.} In math-talk, ``show'' means ``prove''.} \par\noindent{\small {\bf Tip.} Don't start working! Read the whole exam first. You may wish to start with the questions that are easiest for you.} \par\noindent{\small {\bf Tip.} Neatness, cleanliness and organization count, here and everywhere else!} \fullcut \fullcut \noindent{\bf Problem 2.} State and prove the ``dimension theorem'', also known as the ``rank-nullity theorem'', for a given linear transformation $T\colon V\to W$. \vskip 1mm \par\noindent{\small {\bf Tip.} In math-talk, ``state'' means ``write the statement of, in full''.} \par\noindent{\small {\bf Tip.} As always in math exams, when proving a theorem you may freely assume anything that preceded it but you may not assume anything that followed it.} \fullcut \fullcut % (M = {{0, 1, 1}, {1, 0, 1}, {1, 1, 0}}.{{1, -2, 0, -1, 2, -1}, {0, 0, 1, 3, 1, -1}, {0, 0, 0, 0, 1, 0}}) // MatrixForm % (A = Take[M, All, 4]) // MatrixForm % {v1, v2} = Thread[Take[M, All, -2]] % LinearSolve[A, v1] % Solve[A.{x1, x2, x3, x4} == -v2, {x1, x2, x3, x4}] \noindent{\bf Problem 3.} Find all the solutions (if any exist) of the following two systems of linear equations: \[ \begin{pmatrix}0&0&1&3\\1&-2&0&-1\\1&-2&1&2\end{pmatrix} \begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix} = \begin{pmatrix}3\\3\\4\end{pmatrix} \qquad\text{and}\qquad \begin{pmatrix}0&0&1&3\\1&-2&0&-1\\1&-2&1&2\end{pmatrix} \begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix} = \begin{pmatrix}1\\2\\3\end{pmatrix}. \] \vskip 1mm \par\noindent{\small {\bf Tip.} Show all intermediate steps!} \par\noindent{\small {\bf Tip.} It is always an excellent idea to substitute your solutions back into the equations and see if they really work.} \fullcut \fullcut \noindent{\bf Problem 4.} Let $A$ be the matrix $A=\left( \begin{array}{cc} -4 & 6 \\ -3 & 5 \\ \end{array} \right)$. \begin{enumerate} \item Compute $\det(A-\lambda I)$. \item Find the eigenvalues $\lambda_1$ and $\lambda_2$ of $A$. \item Find their corresponding eigenvectors $v_1$ and $v_2$. \item Find a matrix $C$ for which $AC=CD$, where $D=\left( \begin{array}{cc} \lambda_1 & 0 \\ 0 & \lambda_2 \\ \end{array} \right)$. \item Compute the inverse of $C$. \item For an arbitrary natural number $n$, compute $A^n$ by computing $CD^nC^{-1}$. \end{enumerate} \fullcut \fullcut \noindent{\bf Problem 5.} In the spaces provided, indicate for each of the following statements if it is TRUE or FALSE. You may leave an entry blank. You will earn $2$ points for each correct answer, $(-3)$ points for each incorrect answer, and $0$ points for each entry left blank. Your overall mark for this question will be raised to $0$ if you earn a negative number of points. If nothing is otherwise stated, $A$ and $B$ are assumed to be square $n\times n$ matrices with entries in some field $F$. \begin{enumerate}[leftmargin=*,labelindent=16pt,label={\bfseries\arabic*. \underline{\hspace{1in}}}] \item If a row of $A$ is identical to one of the columns of $A$, then $\det(A)=0$. \item The determinant of $A$ can be evaluated using an expansion along any row or column. \item If two rows of $A$ are identical then $\det(A)=0$. \item If $B$ is obtained from $A$ by interchanging any two columns, then $\det(B)=-\det(A)$. \item If $B$ is obtained from a $A$ by multiplying a row of $A$ by a scalar, then $\det(B)=\det(A)$. \item The function $\det\colon M_{n\times n}(F)\to F$ is a linear transformation. \item If $B$ is obtained from $A$ by adding $c$ times one row to another row, then $\det(B)=c\det(A)$. \item If $A$ is an $n\times n$ matrix where $n$ is even, then $\det(A)=\det(-A)$. \item If $A$ has rank $n$, then $\det(A)\neq 0$. \item If $A$ is upper triangular (meaning that $A_{ij}=0$ whenever $i>j$), then $\det(A)$ is equal to the product of the diagonal entries $A_{ii}$ of $A$. \item Always, $\det(A)=\det(A^T)$. \item If $AB\neq BA$ then $\det(AB)\neq\det(BA)$. \item If $A$ is invertible, then $\det(A)=\det(A^{-1})$. \item Always, $\det(AB)=\det(BA)$. \end{enumerate} \vskip 1mm \par\noindent{\small {\bf Tip.} There is no need to justify your answers. \par\noindent{\small {\bf Tip.} Note that a statement is considered TRUE only if it is {\bf always} true, and not just {\bf sometimes}.} \fullcut \fullcut \fullcut \fullcut \fullcut \fullcut \vfill \label{goodluck}\centerline{\bf Good Luck!} \end{document} \def\tfbox{{$\underline{\hspace{1in}}$}} \def\qbox#1{\parbox{4in}{#1}} \begin{tabular}{rcl} 1. & \tfbox & \qbox{The function $\det\colon M_{n\times n}(F)\to F$ is a linear transformation.} \\ 1. & \tfbox & \qbox{The determinant of $A$ can be evaluated using an expansion along any row or column.} \\ 1. & \tfbox & \qbox{If two rows of $A$ are identical then $\det(A)=0$.} \\ 1. & \tfbox & \qbox{If a row of $A$ is identical to one of the columns of $A$, then $\det(A)=0$.} \\ 1. & \tfbox & \qbox{If $AB\neq BA$ then $\det(AB)\neq\det(BA)$.} \\ 1. & \tfbox & \qbox{If $B$ is obtained from $A$ by interchanging any two columns, then $\det(B)=-\det(A)$.} \\ 1. & \tfbox & \qbox{If $B$ is obtained from a $A$ by multiplying a row of $A$ by a scalar, then $\det(B)=\det(A)$.} \\ 1. & \tfbox & \qbox{If $B$ is obtained from $A$ by adding $c$ times one row to another row, then $\det(B)=c\det(A)$.} \\ 1. & \tfbox & \qbox{If $A$ has rank $n$, then $\det(A)\neq 0$.} \\ 1. & \tfbox & \qbox{If $A$ is upper triangular (meaning that $A_{ij}=0$ whenever $i>j$), then $\det(A)$ is equal to the product of the diagonal entries $A_{ii}$ of $A$.} \\ 1. & \tfbox & \qbox{Always, $\det(A)=\det(A^T)$.} \\ 1. & \tfbox & \qbox{If $A$ is invertible, then $\det(A)=\det(A^{-1})$.} \\ 1. & \tfbox & \qbox{If $A$ is an $n\times n$ matrix where $n$ is even, then $\det(A)=\det(-A)$.} \\ 1. & \tfbox & \qbox{Always, $\det(AB)=\det(BA)$.} \end{tabular}