worksheet4.tex

\documentclass{exam}
\usepackage{amsmath}
\usepackage{amsfonts}

\begin{document}

\begin{center}
    Worksheet 4 - Due 10/27
\end{center}
\begin{questions}
    \question
    Find an example of each of the following. If it is not possible, write NOT
    POSSIBLE.
    \begin{parts}
       \part
        Give an example of 2 linear transformations $S,T:\mathbb{R}^3 \to
        \mathbb{R}^3$ (this means they are both from $\mathbb{R}^3$ to
        $\mathbb{R}^3$) such that $S$ is onto but $S\circ T$ (this is the
        function given by $(S\circ T)(x)=S(T(x))$) is not.
        \part
        Give an example of 2 linear transformations $S,T:\mathbb{R}^3 \to
        \mathbb{R}^3$ such that $T$ is onto but $S\circ T$ is not.
        \part
        Give an example of 2 linear transformations $S,T:\mathbb{R}^3 \to
        \mathbb{R}^3$ such that $S$ is one-to-one but $S\circ T$ is not.
        \part
        Give an example of 2 linear transformations $S,T:\mathbb{R}^3 \to
        \mathbb{R}^3$ such that $T$ is one-to-one but $S\circ T$ is not.
    \end{parts}

    \question
    Give a linear transformation $T:\mathbb{R}^2\to\mathbb{R}^2$ such that
    $T(1,1)=(2,3)$ and $T(-1,2)=(0,1)$. Do this using matrix inverses.

    \question
    Find an example of each of the following. If it is not possible, write NOT
    POSSIBLE.
    \begin{parts}
       \part
        Give an example of a linear transformation
        $T:\mathbb{R}^2\to\mathbb{R}^2$ that reflects every point about the
        $x$-axis.
        \part
        Give an example of a linear transformation
        $T:\mathbb{R}^2\to\mathbb{R}^2$ that reflects every point about the
        $x=y$ line.
        \part
        Give an example of a linear transformation
        $T:\mathbb{R}^2\to\mathbb{R}^2$ that shifts every point up by one unit.
    \end{parts}
\end{questions}

\end{document}