Name:      

Directions: Show all work. No credit for answers without work.

1. [2 parts, 2 points each] Decide whether the given transformation is linear. If the transformation is is linear, give the standard matrix. If the transformation is not linear, then explain why.

   1. $\left[\begin{array}{c}\hfill x_1\\ \hfill x_2\end{array}\right]\mapsto \left[\begin{array}{c}\hfill 0\\ \hfill |x_1+x_2|\end{array}\right]$
   2. $\left[\begin{array}{c}\hfill x_1\\ \hfill x_2\end{array}\right]\mapsto \left[\begin{array}{c}\hfill e^3{x}_1+\tan <!--\; FUNCTION\; APPLICATION\; -->(\pi ∕5)x_2\\ \hfill -x_1\end{array}\right]$.

2. [1 point] Let $T:{ℝ}^n\to {ℝ}^m$ be a linear transform, and let ${𝐯}_1,\dots {𝐯}_p$ be vectors in ${ℝ}^n$. Show that if $\{{𝐯}_1,\dots ,{𝐯}_p\}$ is a linearly dependent set, then $\{T({𝐯}_1),\dots ,T({𝐯}_p)\}$ is linearly dependent.

3. [2 parts, 2 points each] Suppose that $T:{ℝ}^2\to {ℝ}^3$ is a linear transform, let $𝐮=\left[\begin{array}{c}\hfill -4\\ \hfill 1\end{array}\right]$ and let $𝐯=\left[\begin{array}{c}\hfill 3\\ \hfill -2\end{array}\right]$. We know that $T$ maps $𝐮$ to $\left[\begin{array}{c}\hfill 1\\ \hfill -2\\ \hfill 4\end{array}\right]$ and $T$ maps $𝐯$ to $\left[\begin{array}{c}\hfill 2\\ \hfill -1\\ \hfill -1\end{array}\right]$.

   1. Find the image of $2𝐮-𝐯$ under $T$.
   2. If possible, then find $T(𝐰)$, where $𝐰=\left[\begin{array}{c}\hfill 1\\ \hfill -19\end{array}\right]$. If not possible, then explain why not.
4. [1 point] Give a simple example of a linear transformation $T:{ℝ}^2\to {ℝ}^3$ such that the range of $T$ is $\left\{\left[\begin{array}{c}\hfill x_1\\ \hfill x_2\\ \hfill x_3\end{array}\right]:x_1+x_2+x_3=0\right\}$.