\documentclass[11pt]{article}
\usepackage{jeffe,handout,multicol,tikz,kgm}
\usetikzlibrary{arrows.meta}

\newcommand{\kgmcomment}[1]{}

\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}

\begin{document}
\headers{Math343}{Quiz 5}{Sep 26, 2025}

%\makeatletter
%\def\@oddfoot{\hfill \tikz \node[draw] {over} ; }
%\makeatother

\makeatletter
\def\@oddfoot{}
\makeatother

\newcommand{\dd}{\partial}

\noindent
\hfill Name: \underline{\hspace{3.5in}} 

%\begin{flushright}
%Name: \underline{\hspace{3.5in}} \\
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%\hrule

\newcommand{\vect}[1]{\mathbf{#1}}
\newcommand{\Span}{\mathrm{Span}}
\newcommand{\RR}{\mathbb{R}}

% Matrix Three By One
\newcommand{\MTBO}[3]%
{%
	\left[%
	\begin{array}{*{3}{r}}
	#1\\
	#2\\
	#3
	\end{array}%
	\right]%
}

% Matrix Three By One
\newcommand{\MTWBO}[2]%
{%
	\left[%
	\begin{array}{*{3}{r}}
	#1\\
	#2
	\end{array}%
	\right]%
}

%\matrix{n}{data}
\newcommand{\mtrix}[2]%
{%
	\left[
	\begin{array}{*{#1}{r}}
		#2
	\end{array}
	\right]
}

\newcommand{\mb}[1]{\mathbf{#1}}
%
% e.g. \kgmarray{num cols}{data}
%
\newcommand{\kgmarray}[2]%
{% 
    \begin{array}{*{#1}{r}}
        #2
    \end{array}
}

%
% e.g. \kgmmatrix{num cols}{data}
%
\newcommand{\kgmmatrix}[2]%
{% 
    \left[ \kgmarray{#1}{#2} \right]
}

%
% e.g. A \rrrule{$R2 \stackrel{+}{=} (-3)R1$ \\ $R3 \stackrel{+}{=} R1$}
%
\newcommand{\rrrule}[1]%
{%
    \begin{tikzpicture}
        \scriptsize
        \node[rectangle,align=left] (TMP) {#1} ;
        \path (TMP.south west) ++(-0.3ex, -0.3ex) node[coordinate] (TMPLEFT) {} ;
        \path (TMP.south east) ++(0.3ex, -0.3ex) node[coordinate] (TMPRIGHT) {} ;
        \draw[->] (TMPLEFT) -- (TMPRIGHT) ;
    \end{tikzpicture}
}      

\tikzset{>={Latex[length=3mm, width=2mm]}}

\setlength{\columnseprule}{0.4pt}

\noindent
\textbf{Directions:} Show all work.  No credit for answers without work.

\begin{enumerate}

\newcommand{\myspc}{\vspace{4cm}\\~}
\item\ppoints{4}{1} Determine whether the following vectors are linearly independent or linearly dependent.  Justify your answer.
\begin{multicols}{2}
\begin{enumerate}
\item $\kgmmatrix{1}{2 \\ 3}, \kgmmatrix{1}{-4 \\ -6}$
\myspc
\item $\kgmmatrix{1}{2 \\ 0 \\ 0 \\ 0}, \kgmmatrix{1}{5 \\ 4 \\ 0 \\ 0}, \kgmmatrix{1}{-1 \\ 5 \\ -7 \\ 0}$
\myspc

\columnbreak
\item $\kgmmatrix{1}{5 \\ 0 \\ 0 \\ 0}, \kgmmatrix{1}{5 \\ 1 \\ 0 \\ 0}, \kgmmatrix{1}{-2 \\ -3 \\ 0 \\ 0}$
\myspc 
\item $\kgmmatrix{1}{1 \\ 2 \\ 3}, \kgmmatrix{1}{4 \\ 5 \\ 6}, \kgmmatrix{1}{7 \\ 8 \\ 9}, \kgmmatrix{1}{10 \\ 11 \\ 12}$
\myspc
\end{enumerate}
\end{multicols}
\item\points{3} Determine the values of $h$ that make the following vectors linearly independent.
\[ \kgmmatrix{1}{2 \\ 14 \\ 2}, \kgmmatrix{1}{-1 \\ h \\ -1}, \kgmmatrix{1}{5 \\ 35 \\ h+1} \]


\newpage

\item\ppoints{2}{1} True/False.  Justify your answers.
\begin{enumerate}
\item If the columns of $A$ are linearly independent, then $A\mb{x} = \mb{0}$ has a unique solution.
\vspace{4cm}
\item If $A\mb{x} = \mb{b}$ has a unique solution for at least one vector $\mb{b}$, then the columns of $A$ are linearly independent.
\vspace{4cm}
\end{enumerate}

\item\points{1} Prove that if $\mb{x}$ and $\mb{y}$ are linearly independent but $\{\mb{x},\mb{y},\mb{z}\}$ is linearly dependent, then $\mb{z}\in\Span\{\mb{x},\mb{y}\}$.
\vfill
\end{enumerate}
\end{document}
