\documentclass[12pt]{article}
\usepackage[box,completemulti]{tabletgrader}
\usepackage{amsmath,amsfonts}
\usepackage{enumerate}
\usepackage{siunitx}
\usepackage{multicol}
\everymath{\displaystyle}
\Course{Math 313}{Linear Algebra}
\Instructor{Steven M. McKay}
\TestName{Exam 1}
\Date{Sep 25,26}
\Stamp{NS}
\Barcode{barcodepic.pdf}
\newcommand{\vct}[1]{\langle #1\rangle}
\newcommand{\vc}[1]{\bf #1}
\newcommand{\R}{\mathbb{R}}
\begin{document}


\element{mcquestions}{
\begin{mcquestion}{A-1-1}
What is the solution to the linear system
\begin{eqnarray*} 2x_1+x_2-x_3&=3\\
x_2+2x_3&=-4\\
3x_3&=-3?\end{eqnarray*}
\begin{choices}
\correctchoice{$(2,-2,-1)$}
\wrongchoice{$(-2,-2,-1)$}
\wrongchoice{$(-2,2,-1)$}
\wrongchoice{$(-2,2,1)$}
\wrongchoice{$(2,2,1)$}
\wrongchoice{$(0,0,0)$}
\end{choices}
\end{mcquestion}

\vspace{1in}
\begin{questionmult}{A-1-2-p1}\scoring{b=1,m=0}
Which of the following statements are not always true: (Indicate {\bfseries all} that apply).
\begin{choices}
\correctchoice{A linear system of equations can have exactly two solutions.}
\wrongchoice{The row echelon form of an inconsistent system of equations has a pivot in the last column.}
\correctchoice{A consistent system has a unique solution.}
\wrongchoice{The row echelon form of a matrix is not unique.}
\wrongchoice{None of the statements are always true.}
\end{choices}
\end{questionmult}
\newpage
}


\element{mcquestions}{
\begin{questionmult}{A-1-2-p2}\scoring{b=1,m=0}
Which of the following matrices are not in row echelon form? (Indicate {\bfseries all} that apply).
\begin{choices}
\wrongchoice{$\begin{bmatrix}1&1&1&1\\0&1&0&1\\0&0&1&0\end{bmatrix}$}
\correctchoice{$\begin{bmatrix}1&0&0&0&\\0&0&0&0\\0&0&1&0\end{bmatrix}$}
\wrongchoice{$\begin{bmatrix}1&1&0&0\\0&1&0&1\\0&0&0&0\end{bmatrix}$}
\correctchoice{$\begin{bmatrix}1&0&0\\0&1&0\\1&0&1\end{bmatrix}$}
\wrongchoice{$\begin{bmatrix}1&1&1\\0&0&0\\0&0&0\end{bmatrix}$}
\end{choices}
\end{questionmult}

\vspace{1in}


\begin{mcquestion}{A-1-3}
Let $\vc v_1=\begin{bmatrix}-1\\ 2\end{bmatrix}$ and $\vc v_2=\begin{bmatrix}2\\ 1\end{bmatrix}.$ If $\vc w=\begin{bmatrix}-4\\3\end{bmatrix}$, then which of the following statements are true?
\begin{choices}
\correctchoice{$2\vc v_1-\vc v_2=\vc w$}
\wrongchoice{$2\vc v_1+\vc v_2=\vc w$}
\wrongchoice{$-2\vc v_1+\vc v_2=\vc w$}
\wrongchoice{$\vc v_1-2\vc v_2=\vc w$}
\wrongchoice{$-\vc v_1+2\vc v_2=\vc w$}
\wrongchoice{$\vc v_1-2\vc v_2=\vc w$}
\end{choices}
\end{mcquestion}
\newpage
}


\element{mcquestions}{
\begin{mcquestion}{A-1-4}
Let 
\[A=\begin{bmatrix}1&0&1\\1&1&0\\0&-1&1\end{bmatrix}.\]
For which of the following vectors $\vc b$ is the system $A\vc x=\vc b$ inconsistent?
\begin{multicols}{2}
\begin{choices}
\correctchoice{$\begin{bmatrix}1\\1\\1\end{bmatrix}$}
\wrongchoice{$\begin{bmatrix}1\\2\\-1\end{bmatrix}$}
\wrongchoice{$\begin{bmatrix}-2\\-3\\1\end{bmatrix}$}
\wrongchoice{$\begin{bmatrix}0\\1\\-1\end{bmatrix}$}
\wrongchoice{$\begin{bmatrix}-1\\0\\-1\end{bmatrix}$}
\wrongchoice{$\begin{bmatrix}0\\0\\0\end{bmatrix}$}
\end{choices}
\end{multicols}
\end{mcquestion}

\vspace{1in}


\begin{questionmult}{A-1-5}\scoring{b=1,m=0}
In this question, $A$ is a $4\times 4$ matrix.Which of the following statements are always true? (Mark all that apply).
\begin{choices}
\correctchoice{If the system $A\vc x=0$ has only the trivial solution, then $A\vc x=\vc b$ is consistent for all values of $\vc b\in \R^4$.}
\wrongchoice{ If the system $A\vc x=0$ has at least one free variable, then the solution describes a line through the origin.}
\correctchoice{If the system $A\vc x=0$ has two pivots, then it has two free variables.}
\wrongchoice{The only solutions to $A\vc x=\vc b$ describe the trivial space $\{\vc 0\}$, lines through the origin, planes through the origin, hyperplanes through the origin, and $\R^4$ itself.}
\correctchoice{If the system $A\vc x=0$ has three free variables, then the rows of $A$ are all multiples of the first row.}
\end{choices}
\end{questionmult}
\newpage
}


\element{mcquestions}{
\begin{mcquestion}{A-1-5-p2}  If $A$ is a $5\times 3$ matrix and has 3 pivots, then which of the following statements are true? (Choose the best answer.)
\begin{choices}
\correctchoice{The system $A\vc x=0$ has a unique solution.}
\wrongchoice{The system $A\vc x=\vc b$ has a unique solution for all choices of $\vc b$.}
\wrongchoice{The system $A\vc x=\vc b$ has at least one free variable.}
\wrongchoice{The system $A\vc x=\vc b$ has more than one solution, but the system $A\vc x=\vc 0$ has only the trivial solution.}
\wrongchoice{All of these statements are true.}
\wrongchoice{None of these statements are true.}
\end{choices}
\end{mcquestion}

\vspace{1in}



\begin{questionmult}{A-1-7}\scoring{b=1,m=0} Which of the following sets are linearly independent? (Mark all that apply).
\begin{multicols}{2}
\begin{choices}
\correctchoice{$\begin{bmatrix}1\\1\\1\end{bmatrix},\ \begin{bmatrix}1\\1\\0\end{bmatrix}$}
\correctchoice{$\begin{bmatrix} 2\\1\\0\end{bmatrix},\ \begin{bmatrix}1\\1\\-1\end{bmatrix},\ \begin{bmatrix}1\\1\\1\end{bmatrix}$}
\wrongchoice{$\begin{bmatrix} 1\\1\\1\end{bmatrix},\  \begin{bmatrix} 2\\ 1\\3\end{bmatrix},\ \begin{bmatrix}0\\0\\0\end{bmatrix}$}
\wrongchoice{$\begin{bmatrix} 2\\1\\1\end{bmatrix},\ \begin{bmatrix} 1\\0\\1\end{bmatrix},\ \begin{bmatrix} -3\\1\\1\end{bmatrix},\ \begin{bmatrix} 2\\4\\2\end{bmatrix}$}
\wrongchoice{$\begin{bmatrix}1\\0\\1\end{bmatrix},\ \begin{bmatrix} 1\\2\\-1\end{bmatrix},\ \begin{bmatrix}2\\ 1\\ 1\end{bmatrix}$}
\end{choices}
\end{multicols}
\end{questionmult}
\newpage
}


\element{mcquestions}{
\begin{mcquestion}{A-1-8} Let
\[A=\begin{bmatrix}1&1&2&2&3\\-1&2&-2&2&4\end{bmatrix}.\]  Which of the following vectors is not in the range of the linear transformation $T(\vc x)=A\vc x$?
\begin{choices}
\wrongchoice{$\begin{bmatrix}1\\1\end{bmatrix}$}
\wrongchoice{$\begin{bmatrix}2\\-1\end{bmatrix}$}
\wrongchoice{$\begin{bmatrix}-4\\3\end{bmatrix}$}
\wrongchoice{$\begin{bmatrix}0\\0\end{bmatrix}$}
\wrongchoice{$\begin{bmatrix}-2\\-2\end{bmatrix}$}
\correctchoice{All of these are in the range.}
\end{choices}
\end{mcquestion}

\vspace{1in}


\begin{mcquestion}{A-1-9} What is the matrix of the transformation $T:\R^2\to \R^2$ that rotates a vector counterclockwise through $\pi/4$ radians and then reflects through the $x_1$ axis?
\begin{multicols}{2}
\begin{choices}
\correctchoice{$\begin{bmatrix} \frac{\sqrt 2}2&- \frac{\sqrt 2}2\\- \frac{\sqrt 2}2&- \frac{\sqrt 2}2\end{bmatrix}$}
\wrongchoice{$\begin{bmatrix} -\frac{\sqrt 2}2&- \frac{\sqrt 2}2\\- \frac{\sqrt 2}2&- \frac{\sqrt 2}2\end{bmatrix}$}
\wrongchoice{$\begin{bmatrix} -\frac{\sqrt 2}2& \frac{\sqrt 2}2\\- \frac{\sqrt 2}2&- \frac{\sqrt 2}2\end{bmatrix}$}
\wrongchoice{$\begin{bmatrix} -\frac{\sqrt 2}2&\frac{\sqrt 2}2\\ \frac{\sqrt 2}2&- \frac{\sqrt 2}2\end{bmatrix}$}
\wrongchoice{$\begin{bmatrix} -\frac{\sqrt 2}2& \frac{\sqrt 2}2\\ \frac{\sqrt 2}2& \frac{\sqrt 2}2\end{bmatrix}$}
\wrongchoice{$\begin{bmatrix} \frac{\sqrt 2}2& \frac{\sqrt 2}2\\- \frac{\sqrt 2}2& \frac{\sqrt 2}2\end{bmatrix}$}
\end{choices}
\end{multicols}
\end{mcquestion}
}
\onecopy{75}{

\frontpage

\noindent{\bfseries Instructions}
\begin{itemize}
\item Do not write on the barcode area at the top of each page, or near the four circles on each page.
\item Mark the correct choice in the multiple choice section. Each multiple question is worth 5 points.
\item Show your work in the free response section.  Correct answers without appropriate work will receive no credit.
\item Work in the free response section must be given in the space provided, or on the back of the sheet the problem is on. If you use the back of the page, make sure you write "work on back" or equivalent.
\item Scientific calculators are allowed.  Graphing calculators are not.  No other electronic device is allowed.
\end{itemize}
\newpage
\setmultiplechoice{5}
\noindent{\bfseries Part 1: Multiple Choice} Check  the correct response.  \\
\vspace{.2in}
\shufflegroup{mcquestions}
\insertgroup{mcquestions}

\noindent {\bfseries Part II:} Free Response.  Show all work for each question in the space provided.  If necessary, write ``work on back'' at the bottom of the space, and continue working on the back of the page. {\bfseries DO NOT} write in the administrative areas.

\begin{freequestion}{B-1-1}{12}  Find all solutions to the system of equations given by
\begin{alignat*}{5} 
x_1&+&x_2&+&2x_3&+&x_4&=&1\\
&&2x_2&+&2x_3&+&2x_4&=&0\\
2x_1&-&2x_2&+&x_3&-&x_4&=&0\\ 
x_1&&&+&x_3&&&=&1
\end{alignat*}
Give the solutions in vector form.  If the solution is dependent, list all solutions.  If the solution is inconsistent, show why.
\end{freequestion}
\newpage{\bfseries This page is intentionally blank.  You may use it, if needed, as extra space for the questions on the other side.}\newpage
\begin{freequestion}{B-1-3}{12} In each of the questions below, the matrix $B$ is given by
\[B=\begin{bmatrix}1&2&3&4&5\\5&4&3&2&1\\6&7&8&9&10\end{bmatrix}\]
\begin{enumerate}[a)] 
\item Calculate the reduced echelon form of $B$.

\vspace{3in}

\item Use the above to find all solutions to $B\vc x=\vc 0$.  Write your answer in vector equation form.

\vspace{1.5in}

\item Prove that $B\vc x=\vc c$ is consistent for all values $c$, or find a value $\vc c$ for which the system is inconsistent.
\end{enumerate}
\end{freequestion}

\newpage{\bfseries This page is intentionally blank.  You may use it, if needed, as extra space for the questions on the other side.}\newpage

\begin{freequestion}{B-1-7}{12}  For each of the sets listed below, determine if the set is linearly independent or linearly dependent.  You {\bfseries must} give a valid reason and show all work.  An answer without work, even if correct, will recieve no credit.

\begin{enumerate}[a)]
\item\[\left\{\begin{bmatrix} 1\\0\\0\\2\end{bmatrix},\ \begin{bmatrix}2\\1\\-1\\2\end{bmatrix},\ \begin{bmatrix}0\\1\\1\\3\end{bmatrix}\right\}\]

\vspace{1in}

\item\[\left\{\begin{bmatrix}1\\2\end{bmatrix},\ \begin{bmatrix}-2\\1\end{bmatrix},\ \begin{bmatrix}1\\1\end{bmatrix}\right\}\]


\vspace{1in}


\item\[\left\{\begin{bmatrix}1\\0\\-1\\0\\1\end{bmatrix},\ \begin{bmatrix}1\\2\\2\\5\\0\end{bmatrix},\ \begin{bmatrix}1\\0\\-1\\0\\1\end{bmatrix},\ \begin{bmatrix}1\\2\\1\\4\\-1\end{bmatrix}\right\}\]
\end{enumerate}
\end{freequestion}
\newpage{\bfseries This page is intentionally blank.  You may use it, if needed, as extra space for the questions on the other side.}\newpage
\begin{freequestion}{B-1-9}{14} Let 
\[F=\begin{bmatrix} 1&1&1\\2&0&-1\\0&2&3\\3&3&3\end{bmatrix}\]
and define a linear transformation $T:\R^3\to\R^4$ by $T(\vc x)=F\vc x$.
\begin{enumerate}
\item Is $T$ one to one? Why or why not?

\vspace{1in}

\item Is $T$ onto? Why or why not?

\vspace{1in}

\item Suppose there is another matrix $S$ so that $T(\vc x)=S\vc x$.  Show that $S=F$.
\end{enumerate}
\end{freequestion}
\newpage{\bfseries This page is intentionally blank.  You may use it, if needed, as extra space for the questions on the other side.}
\clearpage
}
\end{document}
