Activity2.5.2

In this activity, we will look at some examples of matrix transformations.

  1. To begin, suppose that \(A\) is the matrix

    \begin{equation*} A = \left[\begin{array}{rr} 2 \amp 1 \\ 1 \amp 2 \\ \end{array}\right] \text{.} \end{equation*}

    We define the matrix transformation \(T(\xvec) = A\xvec\) so that

    \begin{equation*} T\left(\twovec{-2}{3}\right) = A\twovec{-2}{3} = \left[\begin{array}{rr} 2 \amp 1 \\ 1 \amp 2 \\ \end{array}\right] \twovec{-2}{3} = \twovec{-1}{4} \text{.} \end{equation*}

    The function \(T\) takes the vector \(\twovec{-2}{3}\) as an input and gives us \(\twovec{-1}{4}\) as the output.

    1. What is \(T\left(\twovec{1}{-2}\right)\text{?}\)

    2. What is \(T\left(\twovec{1}{0}\right)\text{?}\)

    3. What is \(T\left(\twovec{0}{1}\right)\text{?}\)

    4. Is there a vector \(\xvec\) such that \(T(\xvec) = \twovec{3}{0}\text{?}\)

  2. Suppose that \(T(\xvec) = A\xvec\) where

    \begin{equation*} A=\left[\begin{array}{rrrr} 3 \amp 3 \amp -2 \amp 1 \\ 0 \amp 2 \amp 1 \amp -3 \\ -2 \amp 1 \amp 4 \amp -4 \end{array}\right] \text{.} \end{equation*}
    1. What is the dimension of the vectors \(\xvec\) that are inputs for \(T\text{?}\)

    2. What is the dimension of the vectors \(T(\xvec)=A\xvec\) that are outputs?

    3. Describe the vectors \(\xvec\) for which \(T(\xvec) = \zerovec\text{.}\)

  3. If \(A\) is the matrix \(A=\left[\begin{array}{rr} \vvec_1 \amp \vvec_2 \end{array}\right]\text{,}\) what is \(T\left(\twovec{0}{1}\right)\) in terms of the vectors \(\vvec_1\) and \(\vvec_2\text{?}\)

  4. Suppose that \(A\) is a \(3\times 2\) matrix and that \(T(\xvec)=A\xvec\text{.}\) If

    \begin{equation*} T\left(\twovec{1}{0}\right) = \threevec{3}{-1}{1}, T\left(\twovec{0}{1}\right) = \threevec{2}{2}{-1} \text{,} \end{equation*}

    what is the matrix \(A\text{?}\)

in-context