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\begin{document}

\title{MATH 351, Fall 2017, Homework \#4}

\author{\underline{\bf Due Friday, September 29}}



\maketitle


\begin{problem}[Exercise 3.5 on Page 158]
	 Find a matrix $A$ such that multiplication by $A$ transforms the unit square $\{(0,0), (1,0), (0,1), (1,1)\}$ onto the parallelogram with corners $\{(0,0), (4,2), (2,4), (6,6)\}$.
\end{problem}

\begin{problem}[Exercise 3.7 on Page 158]
	Find a $2 \times 2$ matrix $B$ such that multiplication by $A$ transforms the parallelogram with corners $$\{(0,0), (2,1), (1,2), (3,3)\}$$ onto the parallelogram with corners $$\{(0,0), (1,0), (1,1), (2,1)\}.$$ How many possible answers does this question admit?
\end{problem}

\begin{problem}[Exercise 3.8 on Page 158]
	Show (using geometric reasoning) that the transformation which reflects points in $\mathbb{R}^2$ on the $x$ axis is linear. Find a matrix that describes this transformation.
\end{problem}

\begin{problem}[Exercise 3.13(a)(b) on Page 160]
	Although it seems silly, it is possible to do elementary row operations on $n \times 1$ matrices. Every such operation defines a transformation of $\mathbb{R}^3$ onto $\mathbb{R}^3$. For example, if we define a transformation of $\mathbb{R}^3$ into $\mathbb{R}^3$ by ``add twice row 1 to row 3,'' this transformation transforms
	$$\begin{bmatrix*}[r] x_1\\ x_2\\ x_3\end{bmatrix*} \rightarrow \begin{bmatrix*}x_1\\ x_2\\ x_3 +2x_1\end{bmatrix*} = \begin{bmatrix*}[r] 1 & 0 & 0\\ 0 & 1 & 0\\ 2 & 0 & 1\end{bmatrix*} \,\begin{bmatrix*}[r] x_1\\ x_2\\ x_3\end{bmatrix*}.$$
	Since this transformation is described by a matrix, we see that our elementary row operation defines a linear transformation. A transformation defined by a single elementary row operation is called an \textbf{elementary transformation} and the matrix that describes such a transformation is called an \textbf{elementary matrix}. \newline
	Find matrices that describe the elementary row operations: (a) ``Add twice row 3 to row 2'' in $\mathbb{R}^4$ and (b) ``Multiply row 2 by 17'' in $\mathbb{R}^3$. 
\end{problem}


\begin{problem}[Exercise 3.35 on Page 174] Define a transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ by the following rule: $T(X)$ is the result of first rotating $X$ counterclockwise by $\frac{\pi}{6}$ radians and then multiplying by 
	$$A =\begin{bmatrix*}[r] 1 & 1\\ 0 & 1\end{bmatrix*}$$
	
\end{problem}


\begin{itemize}
\item[(a)] What is the image of the unit square under $T$? (Rotation is always about the origin when it is considered as matrices unless otherwise specified)
\item[(b)] Find a matrix $B$ such that $T(X) = BX$ for all $X \in \mathbb{R}^2$
\end{itemize}


\begin{problem}[Exercise 3.36 on Page 175] Suppose that in Exercise 3.35 (Problem 5 in this Homework) we multiply first by $A$ and then rotate. Are your answers to parts (a) and (b) different? How?
\end{problem}


\begin{problem}[Exercise 3.44 on Page 175] Find all $2\times 2$ matrices $B$ such that $AB = BA$ where $$A = \begin{bmatrix*}[r] 1 & 2\\0 & 3\end{bmatrix*}.$$
\end{problem}



\begin{problem}[Exercise 3.49 on Page 176] Find a $3 \times 3$ matrix $A$ such that $A^3 = 0$ but $A^2 \neq 0$. [Hint: Try making most of the entries equal to zero.]
\end{problem}


\end{document}


