worksheet4sol.tex

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

\printanswers

\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.
        \begin{solution}
            Let $S$ be the identity and $T$ be the zero transformation.
        \end{solution}
        \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.
        \begin{solution}
            Let $T$ be the identity and $S$ be the zero transformation.
        \end{solution}
        \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.
        \begin{solution}
            Let $S$ be the identity and $T$ be the zero transformation.
        \end{solution}
        \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.
        \begin{solution}
            Let $T$ be the identity and $S$ be the zero transformation.
        \end{solution}
        \part
    \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.
    \begin{solution}
        Let $u_1=(1,1), u_2=(-1,2)$ and $v_1=(2,3),v_2=(0,1)$. Let $U=[u_1 \;
        u_2]$ be the matrix formed by writing $u_1,u_2$ as columns and
        $V=[v_1\; v_2]$ be the matrix formed by writing $v_1,v_2$ as columns.

        Let $F(x)=Ux$ and $G(x)=Vx$. Then we know that $F(e_1)=u_1,F(e_2)=u_2$
        and $G(e_1)=v_1$ and $G(e_2)=v_2$. Since $\{u_1,u_2\}$ is linearly
        indepedent, we know that $U$ and $F$ are invertible. So $G\circ F^{-1}$
        sends $u_1$ to $v_1$ and $u_2$ to $v_2$, as desired. The associated
        matrix to $G\circ F^{-1}$ is $VU^{-1}$.
    \end{solution}

    \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.
        \begin{solution}
            $T(x,y)=(x,-y)$
        \end{solution}
        \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.
        \begin{solution}
            We know that $T(1,1)=(1,1)$ and $T(-1,1)=(1,-1)$. So we can figure
            out what $T$ is by doing the process in question 2.
        \end{solution}
        \part
        Give an example of a linear transformation
        $T:\mathbb{R}^2\to\mathbb{R}^2$ that shifts every point up by one unit.
        \begin{solution}
            NOT POSSIBLE. This function does not map zero to zer.
        \end{solution}
    \end{parts}
\end{questions}

\end{document}