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\begin{document}

\title{Math 18.100B/C, Spring 2012 - Problem Set \# 1}

%\email{jspeck@math.mit.edu}


\maketitle



\fbox{
\begin{minipage}{1 \textwidth}
\textbf{Professors:} Roman Bezrukavnikov and Jared Speck  \ 
\\
\begin{center} 
{\large \textbf{Due: by 3pm on 2-17-12} (in the Undergraduate Math Office: Room 2-108)}
\end{center}
\end{minipage}
}

\bigskip

\textbf{Instructions:} Write your name and whether you are registered for 18.100B or 18.100C. 
Students registered for 18.100C are expected to typeset all of their solutions using LaTex.

\bigskip
\begin{enumerate}

\item Let $\textbf{x} = (x_1,x_2, \cdots, x_k)$ be a vector in $\mathbb{R}^k.$ Let

\begin{align*}
	\textbf{x} \cdot \textbf{y} =  \sum_{i = 1}^k x_i y_i
\end{align*}
denote the \emph{inner product} of $\textbf{x}$ and $\textbf{y},$ 
and let

\begin{align*}
	|\textbf{x}| =  (\textbf{x} \cdot \textbf{x})^{1/2},
\end{align*}
denote the \emph{norm} of $\textbf{x},$ as on pg. 16 of your text. Prove that 

\begin{align*}
	|\textbf{x} \cdot \textbf{y}| = |\textbf{x}| |\textbf{y}|
\end{align*}
if and only if one of the following two possibilities occurs:

\textbf{i)} $\textbf{x} = \textbf{0}$

\textbf{ii)} There exists a real number $c$ such that $\textbf{y} = c\textbf{x}.$

\textbf{Hint:} Take a close look at your book's proof of %the Cauchy-
the Schwarz inequality.

\item Problem 2 from pg. 22.

\item Problem 5 from pg. 22.

\item Problem 6 from pg. 22.

\item Problem 8 from pg. 22.

\item Problem 9 from pg. 22.



\end{enumerate}

\end{document}