###### 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