Activity 2.5.2
In this activity, we will look at some examples of matrix transformations.

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.
What is \(T\left(\twovec{1}{2}\right)\text{?}\)
What is \(T\left(\twovec{1}{0}\right)\text{?}\)
What is \(T\left(\twovec{0}{1}\right)\text{?}\)
Is there a vector \(\xvec\) such that \(T(\xvec) = \twovec{3}{0}\text{?}\)

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*}What is the dimension of the vectors \(\xvec\) that are inputs for \(T\text{?}\)
What is the dimension of the vectors \(T(\xvec)=A\xvec\) that are outputs?
Describe the vectors \(\xvec\) for which \(T(\xvec) = \zerovec\text{.}\)
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 \(\vvec_2\text{?}\)

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 \(A\text{?}\)