Activity 2.5.2

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

1. To begin, suppose that $A$ is the matrix $$A = \left[\begin{array}{rr} 2 & 1 \\ 1 & 2 \\ \end{array}\right].$$ with associated matrix transformation $T(\xvec) = A\xvec$.

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

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

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

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

   5. Write $T\left(\twovec xy\right)$ as a two-dimensional vector.

2. Suppose that $T(\xvec) = A\xvec$ where $$A=\left[\begin{array}{rrrr} 3 & 3 & -2 & 1 \\ 0 & 2 & 1 & -3 \\ -2 & 1 & 4 & -4 \end{array}\right]$$.

   1. What is the dimension of the vectors $\xvec$ that are inputs for $T$?

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

   3. If we describe this transformation as $T:\real^n\to\real^m$, what are the values of $n$ and $m$ and how do they relate to the shape of $A$?

   4. Describe the vectors $\xvec$ for which $T(\xvec) = \zerovec$.

3. If $A$ is the matrix $A=\left[\begin{array}{rr} \vvec_1 & \vvec_2 \end{array}\right]$, what is $T\left(\twovec10\right)$ in terms of the vectors $\vvec_1$ and $\vvec_2$? What about $T\left(\twovec{0}{1}\right)$?

4. Suppose that $A$ is a $3\times 2$ matrix and that $T(\xvec)=A\xvec$. If $$T\left(\twovec{1}{0}\right) = \threevec{3}{-1}{1}, T\left(\twovec{0}{1}\right) = \threevec{2}{2}{-1}$$, what is the matrix $A$?