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\begin{center} {\Large\bf Honors Algebra I, Assignment 1} \\
{\bf Not to Be Submitted}
\end{center}
\noindent Instructor: Prof. Joel Spencer
\\ Office: wwh829
\\ Email: {\tt spencer@cims.nyu.edu}
\\ Phone: x83219 (but email is much better!)
\\ Course website (bookmark!):
\\{\tt www.cs.nyu.edu/cs/faculty/spencer/algebra/index.html}
\\ or go to Prof. Spencer's home page and look
for pointer.
\ben
\item List with description the symmetries of the square. (There
are eight of them.) Give the table for the products of the
symmetries. Give the inverse of each symmetry.
\item Let $G$ denote the set of linear functions $f(x)=mx+b$ on
the real line with $m\neq 0$. Denote such a function by $(m,b)$.
Define a product $f^*g$ as the function $h(x)=g(f(x))$. For
$f(x)=3x+7$ and $g(x)=5x-2$ what are $f^*g$ and $g^*f$? More
generally, for
$f=(m_1,b_1)$ and $g=(m_2,b_2)$ find $f^*g$ and $g^*f$. For
$f=(m,b)$ find a formula for that $g$ so that $f^*g=(1,0)$.
\item Let $GL_n(R)$ denote (this is standard) the general linear
group over $R$. That is, the elements are the $n\times n$
nonsingular matrices $A$ and the group operation is matrix
multiplication. Let $H$ denote those $A\in GL_n(R)$ with
{\em positive} determinant. Prove that $H$ is a subgroup
of $GL_n(R)$.
\\ In general, given a group $G$ (here $GL_n(R)$) and a
nonempty subset $H\subset G$, to prove that $H$ is a
subgroup of $G$ you need to show the following three
things
\ben
\item {\tt Identity:} The identity element $I\in H$.
\item {\tt Multiplicative Closure:} {\em If} $A,B\in H$
then $AB\in H$
\item {\tt Inverse Closure:} {\em If} $A\in H$ then $A^{-1}\in H$.
\een
(Note: In other situations the identity element may have different
names, such as $0,e,1$. Also, if the operation is written as
addition then $AB$ becomes $A+B$ and $A^{-1}$ becomes $-A$.)
\een
\begin{quote}
Beauty is the first test: there is no permanent place in the
world for ugly mathematics. -- G.H. Hardy
\end{quote}
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