\documentclass[12pt]{amsart}

\usepackage{241}

\title{21-241 Matrices and Linear Transformations \\
       Summer I 2012 \\
       Homework 2}

\begin{document}
  \maketitle

  \begin{enumerate}
    \item  Describe the set of solutions to the following system of linear equations, using a particular solution and finitely many solutions to the corresponding homogeneous system. [5]
      \begin{LinearSystems}{5}
        4x - 5y + 2z + w = 1 \\
         &  \-y   &  + w = 0 \\
      \end{LinearSystems}
    \item  Find the inverse of the following matrix: [10] 
      \[
        \begin{pmatrix}
              1       & \frac{1}{2} & \frac{1}{3} \\
          \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\
          \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\
        \end{pmatrix}
      \]
    \item  Let $A$ and $B$ be square matrices of the same size.  Show that if $AB$ is invertible, then so are $A$ and $B$.  [15]
    \item  Prove or disprove; there is a (real) $2\times 2$ matrix $A$ such that $A^2 = -I$.  [5]
    \item  Let $A = \left(\begin{smallmatrix} 1 & 1 \\ 1 & 0 \\ \end{smallmatrix}\right)$.  Show by induction on $n$ that
      \[
        A^n = \begin{pmatrix}
          f_{n+1} & f_n \\
          f_n & f_{n-1} \\
        \end{pmatrix}
      \]
      where $f_n$ is the $n$th Fibonacci number, defined by the recursion $f_0 = 0$, $f_1 = 1$, and $f_{n+2} = f_{n+1} + f_n$.  [10]
    \item  Let $\theta\in [0,2\pi)$, and let $L\subseteq\RR^2$ be the line containing the origin $(0,0)$ which makes an angle of $\theta$ with the $x$-axis.  Let $T : \RR^2\to\RR^2$ be the function which takes some point $x\in\RR^2$ to its reflection across $L$.
      \begin{enumerate}
        \item  Show that $T$ is a linear transformation.  (You can use either algebra or geometry.  If you use geometry, make sure to be unambiguous when drawing your figures, and to illustrate every possible case.)  [10]
        \item  Find the standard matrix of $T$.  [10]
      \end{enumerate}
  \end{enumerate}
\end{document}
