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\documentclass{exam}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\begin{document}
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\begin{center}
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    Worksheet 4 - Due 10/27
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\end{center}
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\begin{questions}
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    \question
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    Find an example of each of the following. If it is not possible, write NOT
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    POSSIBLE.
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    \begin{parts}
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       \part
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        Give an example of 2 linear transformations $S,T:\mathbb{R}^3 \to
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        \mathbb{R}^3$ (this means they are both from $\mathbb{R}^3$ to
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        $\mathbb{R}^3$) such that $S$ is onto but $S\circ T$ (this is the
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        function given by $(S\circ T)(x)=S(T(x))$) is not.
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        \part
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        Give an example of 2 linear transformations $S,T:\mathbb{R}^3 \to
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        \mathbb{R}^3$ such that $T$ is onto but $S\circ T$ is not.
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        \part
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        Give an example of 2 linear transformations $S,T:\mathbb{R}^3 \to
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        \mathbb{R}^3$ such that $S$ is one-to-one but $S\circ T$ is not.
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        \part
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        Give an example of 2 linear transformations $S,T:\mathbb{R}^3 \to
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        \mathbb{R}^3$ such that $T$ is one-to-one but $S\circ T$ is not.
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    \end{parts}
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    \question
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    Give a linear transformation $T:\mathbb{R}^2\to\mathbb{R}^2$ such that
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    $T(1,1)=(2,3)$ and $T(-1,2)=(0,1)$. Do this using matrix inverses.
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    \question
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    Find an example of each of the following. If it is not possible, write NOT
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    POSSIBLE.
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    \begin{parts}
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       \part
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        Give an example of a linear transformation
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        $T:\mathbb{R}^2\to\mathbb{R}^2$ that reflects every point about the
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        $x$-axis.
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        \part
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        Give an example of a linear transformation
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        $T:\mathbb{R}^2\to\mathbb{R}^2$ that reflects every point about the
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        $x=y$ line.
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        \part
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        Give an example of a linear transformation
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        $T:\mathbb{R}^2\to\mathbb{R}^2$ that shifts every point up by one unit.
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    \end{parts}
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\end{questions}
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\end{document}
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