\documentclass[oneside,addpoints,12pt]{exam}
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\firstpageheader{18SU MAT 280}{\em Unit IV Take-Home Exam}{Name:\underline{\hspace{1.75in}}}
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\begin{document}
\begin{center}
\fbox{\fbox{\parbox{6.5in}{
Attempt all parts of this exam. Students are encouraged to use Sage Math and/or a TI calculator to check their work and complete parts of this exam. This exam has \numpoints\, possible points. Scores will be calculated out of \numpoints\, points.
}}}
\end{center}
\begin{questions}
\question[10] Using the word bank provided, fill in the blank to complete the definition of the given word. In these questions, assume that $A$ is a matrix from $\mathbb{R}^{m\times n}$.
\fbox{Word Bank: rank, row space, composition, null space, column space}
\begin{parts}
\part The \wordAnswerLine[row space] of a matrix is a subspace of $\mathbb{R}^n$ spanned by the the rows of $A$.\\[0.25in]
\part The \wordAnswerLine[column space] of a matrix is a subspace of $\mathbb{R}^m$ spanned by the the columns of $A$.\\[0.25in]
\part The \wordAnswerLine[null space] of a matrix is the subspace of $\mathbb{R}^n$ consisting of vectors that are solutions to the equation $A\vec{x}=\vec{0}$.\\[0.25in]
\part The \wordAnswerLine[rank] of a matrix is the number of non-zero rows. Alternately, it is the number of pivot columns of the matrix.\\[0.25in]
\part If $T$ and $S$ are two linear transformation, then $(T\circ S)\left(\vec{v}\right)$ is called the \wordAnswerLine[composition] of two linear transformations.
\end{parts}
\question[10] Box the word \fbox{\Large\textsc{True}} or \fbox{\Large \textsc{False}} to indicate your answer. If the statement is false, explain why or provide an example why the statement is false.
\begin{parts}
\part {\Large\textsc{True} / \textsc{False}}\\
Every plane in $\mathbb{R}^3$ is a subspace of $\mathbb{R}^3$.
\setlength\linefillheight{.28in}
\begin{solutionorlines}[1in]
\end{solutionorlines}\vfill
\part {\Large\textsc{True} / \textsc{False}}\\
The transformation $T:\mathbb{R}^2\to\mathbb{R}^2$ such
that $T\left(\vec{x}\right)=-\vec{x}$ is a linear transformation.
\begin{solutionorlines}[1in]
\end{solutionorlines}\vfill
\end{parts}
\newpage
\question Consider the augmented matrix $\begin{bmatrix}[r|r|r]A & \vec{u} & \vec{v}\end{bmatrix}$ defined below with given reduced row echelon form
\[\begin{bmatrix}[rrrrr|r|r]
4 & 5 & 17 & -5 & 21 & -30 & 2 \\
1 & 5 & 8 & -1 & -10 & -14 & -6 \\
7 & 1 & 22 & 2 & 57 & 6 & 5 \\
5 & -6 & 9 & -5 & 74 & -8 & -2
\end{bmatrix}\quad\stackrel{\mathrm{RREF}}{\widetilde{\qquad}}\quad
\begin{bmatrix}[rrrrr|r|r]
1 & 0 & 3 & 0 & 9 & 0 & 0 \\
0 & 1 & 1 & 0 & -4 & -2 & 0 \\
0 & 0 & 0 & 1 & -1 & 4 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1
\end{bmatrix}
\]
where $A\in\mathbb{R}^{4\times5}$ and $\vec{u},\vec{v}\in\mathbb{R}^4$.
\begin{parts}
\part[6] What is the rank and nullity of $A$?
\begin{solutionorbox}[0.75in]
\end{solutionorbox}
\part[6] Find a basis for $\text{row}(A)$.
\begin{solutionorbox}[1.75in]
\end{solutionorbox}
\part[6] Find a basis for $\text{col}(A)$.
\begin{solutionorbox}[1.75in]
\end{solutionorbox}
\part[8] Find a basis for $\text{null}(A)$.
\begin{solutionorbox}[2.3in]
\end{solutionorbox}
\part[6] Is $\vec{u}$ in the column space of $A$? If so, write $\vec{u}$ as a linear combination of vectors from the basis of $\text{col}(A)$.
\begin{solutionorbox}[1.5in]
\end{solutionorbox}
\part[6] Is $\vec{v}$ in the column space of $A$? If so, write $\vec{v}$ as a linear combination of vectors from the basis of $\text{col}(A)$.
\begin{solutionorbox}[1.5in]
\end{solutionorbox}
\end{parts}
\question[10] Explain (or prove) why the set of all skew-symmetric matrices -- i.e. all matrices that satisfy the equation $A=-A^T$ -- form a subspace of $\mathbb{R}^{n\times n}$.
\begin{solutionorbox}[1.75in]
\end{solutionorbox}
\question[4] Why is the empty set (the set with nothing in it) NOT a subspace?
\begin{solutionorlines}[1in]
\end{solutionorlines}
\question[6] Justify why the transformation $T\left(\begin{bmatrix}x \\ y\end{bmatrix}\right)=\begin{bmatrix}x \\ x+1\end{bmatrix}$ is NOT a linear transformation.
\begin{solutionorbox}[1in]
\end{solutionorbox}
\question[10] Prove that the transformation $T\left(\begin{bmatrix}x \\ y\end{bmatrix}\right)=\begin{bmatrix}2x+y\\y\end{bmatrix}$ is linear.
\begin{solutionorbox}[3.5in]
\end{solutionorbox}
\question Let $T_1$, $T_2$, $T_3$, and $T_4$ be the transformations described below:
\begin{itemize}
\item $T_1$: rotation by $90^\circ$ (counterclockwise),
\item $T_2$: projection onto the $x$-axis,
\item $T_3$: reflection about the line $y=x$,
\item $T_4$: reflection about the $x$-axis.
\end{itemize}
You may use the applet that we built in class for this problem: \url{https://sagecell.sagemath.org/?q=mtduir}.
\begin{parts}
\part[8] Write a matrix for each of the given transformations and label them correctly.
\begin{solutionorbox}[1.5in]
\end{solutionorbox}
\part[4] Find the matrix that first reflects about the $x$-axis and then rotates $90^\circ$ (counterclockwise).
\begin{solutionorbox}[1.5in]
\end{solutionorbox}
\end{parts}
\end{questions}
\end{document}
