Shared18SU / Exams / Unit IV Take-Home Exam / Unit IV Take-Home Exam.texOpen in CoCalc
\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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\makeatletter
where $A\in\mathbb{R}^{4\times5}$ and $\vec{u},\vec{v}\in\mathbb{R}^4$.

%  Link to matrix and solution: https://sagecell.sagemath.org/?q=jxdtpf

\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}