worksheet5.tex

\documentclass{exam}
\usepackage{amsmath}
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\begin{document}

\begin{center}
    Worksheet 5 - Due 11/3
\end{center}

\begin{questions}
    \question
    Extend $\{(1,-1,0,0),(1,0,-1,0)\}$ to a basis for the subspace, $W$, defined by
    $w+x+y+z=0$. In other words, find a basis for $W$ that includes
    $(1,-1,0,0)$ and $(1,0,-1,0)$.

    \question
    Let $P$ be the plane given by $2x+y+z=0$ in $\mathbb{R}^3$.
    \begin{parts}
        \part
        What is a normal vector to $P$?
        \part
        Give a basis for $\mathbb{R}^3$ that includes a normal vector to $P$
        and 2 vectors that lie on $P$.
        \part
        Let $T:\mathbb{R}^3\to\mathbb{R}^3$ be the linear transform that
        reflects all vectors across $P$. This means that $T(n)=-n$ whenver $n$
        is normal to $P$ and $T(v)=v$ if $v$ lies on $P$. Find $A$ such that
        $T(x)=Ax$.
        \part
        What is the rank of $T$? What is the nullity of $T$?
    \end{parts}

    \question
    Let $T:\mathbb{R}^3 \to \mathbb{R}^2$ be the linear transform defined by
    $T(1,1,1)=(1,0)$, $T(1,0,1)=(1,1)$, and $T(1,1,0)=(0,2)$.
    \begin{parts}
        \part
        Before doing a single computation, what can you already say about the
        rank and nullity of $T$?
        \part
        Give a matrix $A$ such that $T(x)=Ax$. You may express $A$ as a product
        of matrices and their inverses.
        \part
        What is the rank and nullity of $T$?
    \end{parts}

    \question
    Give an example of each of the following. If it is not possible, write NOT
    POSSIBLE.
    \begin{parts}
        \part
        Find an invertible $3\times 3$ matrix $A$ and a $3\times 3$ matrix $B$
        such that $rank(AB)\neq rank(BA)$.
        \part
        Find two $3\times 3$ matrices $A$ and $B$, each with nullity 1 such
        that $AB$ is the zero matrix.
        \part
        Find two $3\times 3$ matrices $A$ and $B$, each with rank 1 such
        that $AB$ is the zero matrix.
        \part
        Find two $3\times 3$ matrices $A$ and $B$, each with nullity 2 such
        that $AB$ is the zero matrix.
        \part
        Find two $3\times 3$ matrices $A$ and $B$, each with rank 2 such
        that $AB$ is the zero matrix.
    \end{parts}
\end{questions}

\end{document}