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\lhead{Math 4441 Activity 1}
\rhead{<your name(s)>}


\begin{document}
\title{Math 4441 Activity 1}
\author{<your name(s)>}
\date{August 31, 2022}
\maketitle
\vspace{5em}

\setcounter{section}{1}

\begin{exercise}
Which of the following collections of vectors gives a basis for $\mathbb{R}^3$?  Check the definition for each collection.
\[
\left\{
\left(\begin{matrix} 1\\ 0\\ 0 \end{matrix}\right),
\left(\begin{matrix} 1\\ 1\\ 0 \end{matrix}\right),
\left(\begin{matrix} 1\\ 1\\ 1 \end{matrix}\right),
\left(\begin{matrix} 1\\ 0\\ 1 \end{matrix}\right)
\right\},
\quad
\left\{
\left(\begin{matrix} 1\\ 0\\ 0 \end{matrix}\right),
\left(\begin{matrix} 1\\ 1\\ 0 \end{matrix}\right),
\left(\begin{matrix} 1\\ 1\\ 1 \end{matrix}\right)
\right\},
\quad
\left\{
\left(\begin{matrix} 1\\ 0\\ 0 \end{matrix}\right),
\left(\begin{matrix} 1\\ 1\\ 0 \end{matrix}\right),
\left(\begin{matrix} 1\\ 1\\ 0 \end{matrix}\right)
\right\}
\]
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Explain how the definition of $\measuredangle(\vec{v},\vec{w})$ relies on the Cauchy-Schwarz inequality $\vec{v}\cdot\vec{w} \leq \|\vec{v}\|\,\|\vec{w}\|$.  (One sentence should suffice.)
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Construct an orthonormal basis for $\mathbb{R}^3$ whose first vector is parallel to $\langle 1,1,1 \rangle$.
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Prove that the columns of an orthogonal matrix form an orthonormal basis of $\mathbb{R}^n$.  Also prove that the transpose of an orthogonal matrix is orthogonal.
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Prove that if $A$ is an orthogonal matrix, then $(A\vec{x})\cdot(A\vec{y})=\vec{x}\cdot\vec{y}$, for any vectors $\vec{x},\vec{y}$ in $\mathbb{R}^n$.
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Prove that each of the following functions $\mathbb{R}^3\to\mathbb{R}^3$ are isometries:
\begin{itemize}
	\item $f(\vec{v})=A\vec{v}$, where $A$ is an orthogonal $3\times 3$ matrix;
	\item $g(\vec{v})=\vec{v}+\vec{c}$, where $\vec{c}$ is a fixed vector in $\mathbb{R}^3$.
\end{itemize}
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Expand the above expressions for $\|g(\vec{v})-g(\vec{w})\|^2$ to prove that $g(\vec{v})\cdot g(\vec{w}) = \vec{v}\cdot\vec{w}$, for any vectors $\vec{v},\vec{w}$ in $\mathbb{R}^3$.
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\begin{exercise}
Check that $\{\vec{v}_1,\vec{v}_2,\vec{v}_3\}$ is an orthonormal basis for $\mathbb{R}^3$.
\end{exercise}

\begin{solution}
<your solution>
\end{solution}

\end{document}