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Directions: You may work to solve these problems in groups, but all written work must be your own. Show all work; no credit for solutions without work..

1. [1.8.{13-16}] Use a rectangular coordinate system to plot $𝐮=\left[\begin{array}{c}\hfill 5\\ \hfill 2\end{array}\right]$, $𝐯=\left[\begin{array}{c}\hfill -2\\ \hfill 4\end{array}\right]$ and their images under the given transformation $T$. (Make a separate sketch for each.) Describe geometrically what $T$ does to each vector in ${ℝ}^2$.

   1. $T(𝐱)=\left[\begin{array}{cc}\hfill -1& \hfill 0\\ \hfill 0& \hfill -1\end{array}\right]\left[\begin{array}{c}\hfill x_1\\ \hfill x_2\end{array}\right]$
   2. $T(𝐱)=\left[\begin{array}{cc}\hfill 0.5& \hfill 0\\ \hfill 0& \hfill 0.5\end{array}\right]\left[\begin{array}{c}\hfill x_1\\ \hfill x_2\end{array}\right]$
   3. $T(𝐱)=\left[\begin{array}{cc}\hfill 0& \hfill 0\\ \hfill 0& \hfill 1\end{array}\right]\left[\begin{array}{c}\hfill x_1\\ \hfill x_2\end{array}\right]$
   4. $T(𝐱)=\left[\begin{array}{cc}\hfill 0& \hfill 1\\ \hfill 1& \hfill 0\end{array}\right]\left[\begin{array}{c}\hfill x_1\\ \hfill x_2\end{array}\right]$
2. [1.8.17] Let $T:{ℝ}^2\to {ℝ}^2$ be a linear transformation that maps $𝐮=\left[\begin{array}{c}\hfill 5\\ \hfill 2\end{array}\right]$ to $\left[\begin{array}{c}\hfill 2\\ \hfill 1\end{array}\right]$ and maps $𝐯=\left[\begin{array}{c}\hfill 1\\ \hfill 3\end{array}\right]$ to $\left[\begin{array}{c}\hfill -1\\ \hfill 3\end{array}\right]$. Use the fact that $T$ is linear to find the images under $T$ of $3𝐮$, $2𝐯$, and $3𝐮+2𝐯$.
3. [1.8.32] Suppose vectors ${𝐯}_1,\dots ,{𝐯}_p$ span ${ℝ}^n$, and let $T:{ℝ}^n\to {ℝ}^n$ be a linear transformation. Suppose $T({𝐯}_{𝐢})=0$ for $1\le i\le p$. Show that $T$ is the zero transformation (i.e. $T(𝐱)=0$ for each $𝐱\in {ℝ}^n$).

4. [1.8.{40,41}] Show that the following transformations are not linear.

   1. $\left[\begin{array}{c}\hfill x_1\\ \hfill x_2\end{array}\right]\mapsto \left[\begin{array}{c}\hfill 4x_1-2x_2\\ \hfill 3|x_2|\end{array}\right]$.
   2. $\left[\begin{array}{c}\hfill x_1\\ \hfill x_2\end{array}\right]\mapsto \left[\begin{array}{c}\hfill 2x_1-3x_2\\ \hfill x_1+4\\ \hfill 5x_2\end{array}\right]$.

5. [1.9] True/False. Justify your answer.

   1. A linear transformation $T:{ℝ}^n\to {ℝ}^m$ is completely determined by its effect on the columns of the $n\times n$ identity matrix.
   2. If $T:{ℝ}^2\to {ℝ}^2$ rotates vectors about the origin through an angle $\varphi$, then $T$ is a linear transformation.
   3. The columns of the standard matrix for a linear transformation from ${ℝ}^n$ to ${ℝ}^m$ are the images of the columns of the $n\times n$ identity matrix.
   4. When two linear transformations are performed one after another, the combined effect may not always be a linear transformation.
   5. Not every linear transformation from ${ℝ}^n$ to ${ℝ}^m$ is a matrix transformation.