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\begin{document}

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\vspace{2cm}

\noindent
\textbf{Directions:}  
\begin{enumerate}
	\item Write your name with one character in each box below.
	\item Show all work.  No credit for answers without work.
	\item You are permitted to use one 8.5 inch by 11 inch sheet of prepared notes.  No other aides are allowed.
\end{enumerate}

\newpage
\headers{Math 343}{Test 2}{Oct. 17, 2025}

~
\vspace{1.5cm}
\begin{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Page 1  Linear Independence, Transformations  25 pts
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item\points{15} Determine whether the following vectors are linearly independent.  If the vectors are linearly dependent, give a dependence relation.
\[  \kgmmatrix{1}{1 \\ -1 \\ 2 \\ 3},
	\kgmmatrix{1}{7 \\ -9 \\ -6 \\ 9},
	\kgmmatrix{1}{-3\\ 0 \\ 5 \\ -1},
	\kgmmatrix{1}{0 \\ 4 \\ -1 \\ -2}
\]
\vfill

\item\points{10} Characterize when the following vectors are linearly dependent in terms of simple conditions on $h$ and $k$.
\[  \kgmmatrix{1}{2 \\ 0 \\ 0 \\ 0 \\ 0},
	\kgmmatrix{1}{4 \\ 6 \\ h \\ 0 \\ 0},
	\kgmmatrix{1}{3 \\ k \\ -h \\0 \\ 0},
	\kgmmatrix{1}{1 \\ 5 \\ 1 \\ k \\ 0}
\]
\vfill

\newpage
~
\vspace{1.5cm}
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% Page 2--  Linear Transforms
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item Transformations from $\R^2$ to $\R^2$.
\begin{enumerate}
	\item\points{9} Let $T_1$ be the transform given by $\kgmmatrix{1}{x_1 \\ x_2} \mapsto \kgmmatrix{1}{0 \\ x_2 + 1}$.  Is $T_1$ one-to-one/injective?  Is $T_1$ onto/surjective?  Is $T_1$ linear?  Explain.
	\vfill
	
	\item\points{6} Let $T_2$ be the transform given by $\kgmmatrix{1}{x_1 \\ x_2} \mapsto \kgmmatrix{1}{x_1 + x_2 \\ x_2}$ and let $T_3$ be the transformation that rotates points by $\pi/6$ radians.  Find the standard matrix for the composition transformation $T_2\circ T_3$ given by $\vv x \mapsto T_2(T_3(\vv x))$.
	\vfill
\end{enumerate}

\item Let $T\st \R^n\to\R^m$ be a linear transform, and let $A$ be the standard matrix for $T$.
\begin{enumerate}
	\item\points{2} How many rows does $A$ have?  How many columns?
	\vspace{1cm}
	\item\points{8} By analyzing the pivot positions of $A$, prove that if $n>m$ then $T$ is not one-to-one/injective.
	\vfill
\end{enumerate}


\newpage
~
\vspace{1.5cm}
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% Page 3... 30 pts
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item\points{20} Find the inverse of the following matrix.
\[
\kgmmatrix{3}{
	-8	& 1	& -9 \\
	3	& 0	& 4 \\
	1	& 0	& 1
}
\]
\vfill

\item\points{10} Find elementary matrices $E_1,E_2,E_3$ such that $E_3 E_2 E_1 A = B$.
\begin{align*}
A &= \kgmmatrix{3}{
	1 & 2 & 7 \\
	1 & 2 & 8 \\
	} &
B &= \kgmmatrix{3}{
	1 & 2 & 0 \\
	2 & 4 & 1
}
\end{align*}
\vspace{3cm}



\newpage
~
\vspace{1.5cm}
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% Page 4.. 20 points
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\item\ppoints{10}{2} True/False.  Assume the matrix operations below are well-defined.  Justify your answers.
%\newcommand{\myspc}{\vspace{3cm}\\~}
\newcommand{\myspc}{\vfill}
%\begin{multicols}{2}
\begin{enumerate}
	\item Every elementary matrix is square.
	\myspc 
	\item $(A+B)C = AC+BC$
	\myspc 
	\item $(A+B)(A+B) = A^2 + 2AB + B^2$
	\myspc
	%\item If $AB$ is the zero matrix $0$, then $A=0$ or $B=0$.
	%\myspc
	\item If $AB=BA$, then $A$ and $B$ are inverses of one another.
	\myspc
	\item If $A$ and $B$ are invertible $(n\times n)$-matrices, then $A$ and $B$ are row-equivalent.
	\myspc
	\item If $A$ and $B$ are invertible $(n\times n)$-matrices, then $A+B$ is also invertible.
	\myspc
	\item If $A$ and $B$ are invertible $(n\times n)$-matrices, then $AB$ is also invertible.
	\myspc	
	\item An $(n\times n)$-matrix $A$ is invertible if and only if its transpose $A^T$ is invertible.
	\myspc
	\item If $S$ and $T$ are linear transforms from $\RR^n$ to $\RR^n$ and $S$ and $T$ are equal on at least $n$ points in $\RR^n$, then $S=T$.
	\myspc
	\item If $T$ is a linear transform and $\{\mb v_1,\ldots, \mb v_k\}$ is linearly independent, then $\{T(\mb v_1), \ldots, T(\mb v_k)\}$ is also linearly independent.
	\myspc
\end{enumerate}
%\end{multicols}
\end{enumerate}
\newpage
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