#LyX 1.2 created this file. For more info see http://www.lyx.org/
\lyxformat 220
\textclass article
\language english
\inputencoding latin1
\fontscheme times
\graphics default
\paperfontsize 11
\spacing single 
\papersize Default
\paperpackage a4
\use_geometry 1
\use_amsmath 1
\use_natbib 0
\use_numerical_citations 0
\paperorientation portrait
\leftmargin 1in
\topmargin 1in
\rightmargin 1in
\bottommargin 1in
\secnumdepth 3
\tocdepth 3
\paragraph_separation indent
\defskip medskip
\quotes_language english
\quotes_times 2
\papercolumns 1
\papersides 1
\paperpagestyle default

\layout Section*

Math 314H Projectlet 1: The Art of Proof
\layout Standard


\begin_inset ERT
status Collapsed

\layout Standard

\backslash 
vspace{.1in}
\end_inset 


\layout Standard
\noindent 

\noun on 
Due Date: Monday, March 3, 2003 
\layout Standard
\noindent 

\noun on 
Points: 20 
\layout Standard


\begin_inset ERT
status Collapsed

\layout Standard

\backslash 
vspace{.1in}
\end_inset 


\layout Standard


\series bold 
About Project assignments:
\series default 
 Think of a projectlet as more than an exercise but less than a full blown
 project -- a kind of mini-project.
 It is to be carried out by teams of one or two individuals.
 Start by reading the assignment.
 For this particular assignment, you are not to consult with anyone other
 than your team partner and/or your instructor.
 
\layout Standard

This assignment should be typed up in some document preparation system such
 as WordPerfect, MS Word, Latex, etc., or 
\emph on 
very
\emph default 
 neatly handwritten.
 Check your grammar and spelling (use a spellchecker if you have one.) Like
 any good writing exercise, projectlets should have, as Aristotle advises,
 a beginning (introduction), middle (main body of work) and end (conclusion);
 these parts can be as small as a paragraph.
 Remember that part of your grade will be based on the quality of your written
 work.
 The paper you turn in should be a mix of equations, formulas and prose.
 You should write your answers in such a way that it can be read and understood
 by anyone who knows the material for this course.
 Your write-up should be self-contained and addressed to classmates, not
 your instructor.
 You should cite references and acknowledge all sources in your write-up.
 
\layout Standard


\series bold 
The Problems: 
\series default 
We all know that questions about proving or disproving a mathematical assertion
 can be hard.
 It would be nice to have some advice about how to approach such problems.
 This assignment asks you to prove some results and offer some explanation
 of your own thought processes in arriving at these proofs.
 Write out your proofs using complete sentences and clear justifications
 for steps in your arguments.
 If you need to use other facts about matrices that are in the text, clearly
 reference them.
 Conclude with comments that might be helpful to others regarding how you
 arrived at these proofs.
\layout Standard

This set of facts is centered around the set (actually, vector space) 
\begin_inset Formula $\mathbb{R}^{n\times n}$
\end_inset 

 of 
\begin_inset Formula $n\times n$
\end_inset 

 square matrices with real entries.
 Think of it as a small essay about symmetric and skew-symmetric matrices.
\layout Enumerate

If 
\begin_inset Formula $A$
\end_inset 

 and 
\begin_inset Formula $B$
\end_inset 

 are matrices of the same size, then 
\begin_inset Formula \[
\left(A+B\right)^{T}=A^{T}+B^{T}.\]

\end_inset 


\layout Enumerate

If 
\begin_inset Formula $A$
\end_inset 

 is any square matrix, then 
\begin_inset Formula $\frac{1}{2}\left(A+A^{T}\right)$
\end_inset 

 is symmetric and 
\begin_inset Formula $\frac{1}{2}\left(A-A^{T}\right)$
\end_inset 

 is skew-symmetric.
\layout Enumerate

Every square matrix can be expressed uniquely as the sum of a symmetric
 matrix and a skew-symmetric matrix.
\layout Enumerate

The sets 
\begin_inset Formula $S$
\end_inset 

 of all symmetric matrices and 
\begin_inset Formula $K$
\end_inset 

 of all skew-symmetric matrices are subspaces of the vector space 
\begin_inset Formula $\mathbb{R}^{n\times n}$
\end_inset 

 such that 
\begin_inset Formula $\mathbb{R}^{n\times n}=S+K$
\end_inset 

 and 
\begin_inset Formula $S\bigcap K=\left\{ 0\right\} $
\end_inset 

.
\layout Enumerate


\begin_inset Formula $\dim \mathbb{R}^{n\times n}=\dim S+\dim K$
\end_inset 

.
\layout Standard

If you get stuck, feel free to drop by and see me.
 This assignment can be completed in two to four pages.
\the_end