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\begin{document}
\section*{Lior Silberman's Math 223: Problem Set 1 (due 12/9/12)}
\begin{itemize}
\item Recommended practice problems are from the textbook by Friedberg,
Insel and Spence. They are not for submission.
\item Only numbered problems are for submissions; a problem with {*} or
{*}{*} may be unusually difficult. For your convenience problems taken
from that textbook above are so noted: (\S1.3 E3,4) are problems
3,4 after section 1.3 of that book. RMK indicates a remark, not an
exercise.
\item Lettered problems, as well as problems or subproblems labeled SUPP,
are \textbf{supplementary} and \uline{not for submission;} these
generally cover additional ideas \uline{beyond the scope of the
course}.
\end{itemize}
\begin{center}
\textbf{Practice problems (recommended, but do not submit)}
\par\end{center}
Section 1.2, problems 1-4, 8, 12-13, 17-19.
Section 1.3, problems 1-4, 8, 11, 16-17.
\begin{center}
\textbf{Linear equations}
\par\end{center}
\begin{lyxlist}{10.}
\item [{1.}] Find all solutions in real numbers to the following equations:
(a) $5x+7=13$\\
(b) $\begin{cases}
5x+2y=3\\
6x+4y=2
\end{cases}$ \quad{}(c) $\begin{cases}
3x+2y=a\\
6x+4y=a+1
\end{cases}$ (your answer may depend on the parameter $a$).\bigskip{}
\item [{2.}] In each of the following problems (1) Convert the identity
to a system of linear equations; (2) either exhibit a solution so
that the identity holds true (in which case no proof is needed) or
prove that no such solution exists.
\begin{lyxlist}{10.}
\item [{(a)}] $a\left(x^{2}+2x+1\right)+b\left(5x+3\right)+c\left(2\right)=7x^{2}-5x+3$;
\item [{(b)}] $a\left(x^{2}-2x+1\right)+b\left(x-1\right)+c\left(x^{2}+5x\right)=x^{2}+2x+3$;
\item [{(c)}] $a\left(x^{2}-2x+1\right)+b\left(x-1\right)+c\left(x^{2}-x\right)=x^{2}+2x+3$.\bigskip{}
\end{lyxlist}
\item [{3.}] A matrix $A\in M_{n}(\R)$ is called \emph{skew-symmetric}
if $A^{t}=-A$. Show that $A-A^{t}$ is skew-symmetric for all $A\in M_{n}(\R)$.
You may use the results of problems (\S1.3 E3,4) if you wish.\bigskip{}
\end{lyxlist}
\begin{center}
\textbf{Subspaces}
\par\end{center}
\begin{lyxlist}{10.}
\item [{4.}] In each case decide if the set is a subspace of the given
space.
\begin{lyxlist}{10.}
\item [{(a)}] $V_{1}=\left\{ f\in\R^{\R}\mid\forall t\neq0:f(t)=2f(2t)\right\} $,
$V_{2}=\left\{ f\in\R^{\R}\mid\forall t\neq0:f(t)=f(2t)+1\right\} $
in $\R^{\R}$.
\item [{(b)}] Let $U_{1}=\left\{ \underline{x}\in\R^{3}\mid x_{1}+x_{3}-1=0\right\} $,
$U_{2}=\left\{ \underline{x}\in\R^{3}\mid x_{1}-2x_{2}+x_{3}=0\right\} $,\\
$U_{3}=\left\{ \underline{x}\in\R^{3}\mid x_{1}^{2}+x_{2}^{2}-x_{3}^{2}=0\right\} $
in $\R^{3}$.\bigskip{}
\end{lyxlist}
\item [{5.}] Fix a vector space $V$.
\begin{lyxlist}{10.}
\item [{(a)}] Let $W\subset V$ be a subset. Show that $W$ is a subspace
of $V$ if and only if the following two conditions hold:
\begin{enumerate}
\item $\zv\in W$
\item For all $\vu,\vv\in W$ and $a,b\in\R$ we have $a\vu+b\vv\in W$.
\end{enumerate}
\item [{(b)}] Now let $W\subset V$ be a subspace. For any $n\geq0$ let
$\left\{ \vw_{i}\right\} _{i=1}^{n}\subset W$ be some vectors and
let $\left\{ a_{i}\right\} _{i=1}^{n}\subset\R$ be some scalars.
Give an informal argument showing $\sum_{i=1}^{n}a_{i}\vw_{i}=a_{1}\vw_{1}+\cdots+a_{n}\vw_{n}\in W$.
\item [{BONUS}] Give a formal proof by induction on $n$.
\item [{RMK}] The last item is intended as a diagnostic to see how many
participants can write a proof by induction. \bigskip{}
\end{lyxlist}
\end{lyxlist}
\newpage{}
\begin{lyxlist}{10.}
\item [{6.}] (A chain of subspaces)
\begin{lyxlist}{10.}
\item [{(a)}] Show that the space of bounded functions on a set $X$,
\[
\ell^{\infty}(X)=\left\{ f\in\R^{X}\mid\textrm{There is }M\in\R\textrm{ so that for all }x\in X,\textrm{ we have}\left|f(x)\right|\leq M\right\} \,,
\]
is a subspace of $\R^{X}$.
\item [{(b)}] State (or reconstruct) theorems from calculus to the effect
that ``the space of convergent sequences, $c=\left\{ \underline{a}\in\R^{\N}\mid\lim_{n\to\infty}a_{n}\textrm{ exists}\right\} $,
is a subspace of $\ell^{\infty}(\N)$''.
\item [{RMK}] If you haven't seen those theorems before you can write them
down first and then confirm their existence in your calculus textbook
or Wikipedia. Don't forget that subspaces are subsets!
\item [{(c)}] Show that the space of sequences of finite support, $\R^{\oplus\N}=\left\{ \underline{a}\in\R^{\N}\mid a_{i}\neq0\textrm{ for finitely many }i\right\} $,
is a subspace of $c$. {[}now you need to know a little about convergent
sequences{]}\bigskip{}
\end{lyxlist}
\item [{{*}{*}7.}] (\S1.3 E19) Let $V$ be a vector space and let $W_{1},W_{2}$
be subspaces of $V$. Suppose that union $W_{1}\cup W_{2}=\left\{ v\in V\mid v\in W_{1}\textrm{ or }v\in W_{2}\right\} $
is a subspace of $V$ (note that ``or'' includes the possibility
that both assertions hold). Show that $W_{1}\subset W_{2}$ or $W_{2}\subset W_{1}$.\bigskip{}
\end{lyxlist}
\begin{center}
\textbf{New spaces from old ones}
\par\end{center}
\begin{lyxlist}{10.}
\item [{8.}] Let $V,W$ be two vector spaces. On the set of pairs $V\times W=\left\{ (\vv,\vw)\mid\vv\in V,\vw\in W\right\} $
define $\left(\vv_{1},\vw_{1}\right)+\left(\vv_{2},\vw_{2}\right)=\left(\vv_{1}+_{V}\vv_{2},\vw_{1}+_{W}\vw_{2}\right)$
and $a\cdot\left(\vv_{1},\vw_{1}\right)=\left(a\cdot_{V}\vv_{1},a\cdot_{W}\vw_{1}\right)$.
Show that this endows $V\times W$ with the structure of a vector
space. We will call this space the \emph{external direct sum} of $V,W$
and denote it $V\oplus W$.\bigskip{}
\item [{9.}] Let $W_{1},W_{2}$ be two subspaces of a vector space $V$.
\begin{lyxlist}{10.}
\item [{(a)}] Define their \emph{internal sum} to be the set $W_{1}+W_{2}\eqdef\left\{ \vw_{1}+\vw_{2}\mid\vw_{i}\in W_{i}\right\} $.
Show that $W_{1}+W_{2}$ is a subspace of $V$.
\item [{({*}b)}] Show that $W_{1}\cap W_{2}=\left\{ \zv\right\} $ if and
only if every vector in $W_{1}+W_{2}$ has a \emph{unique} representation
in the form $\vw_{1}+\vw_{2}$.\bigskip{}
\end{lyxlist}
\item [{RMK}] In the case the equivalent conditions of (b) hold, we say
that $W_{1}+W_{2}$ is the \emph{internal direct sum} of $W_{1},W_{2}$
and confusingly also denote this space $W_{1}\oplus W_{2}$. We will
show later that in this case the two ``direct sums'' produced by
problems 8 and 9(b) are in some sense the same. In general it will
be possible to tell from context which direct sum is intended.\bigskip{}
\end{lyxlist}
\newpage{}
\bigskip{}
\begin{center}
\textbf{Supplementary problems: abstractions}
\par\end{center}
\begin{lyxlist}{10.}
\item [{A.}] Write $B^{A}$ for the set of all functions from the set $A$
to the set $B$.
\begin{lyxlist}{10.}
\item [{(a)}] Let $a'$ not be an element of $A$, and let $A'=A\cup\left\{ a'\right\} $
be the set you get by adding $a'$ to $A$. Construct a bijection
between $B^{A'}$ and the set of pairs $B^{A}\times B=\left\{ \left(f,b\right)\mid f\in B^{A},\,b\in B\right\} $.
\item [{(b)}] Suppose that $A,B$ are finite sets. Show that $\#\left(B^{A}\right)=\left(\#B\right)^{(\#A)}$
where $\#X$ denotes the number of elements of a set $X$ and on the
right we have exponentiation of natural numbers.\\
\emph{Hint}: Induction on $\#A$.
\item [{RMK}] Make sure to account for the corner cases where at least
one the sets $A,B$ is empty!\bigskip{}
\end{lyxlist}
\item [{B.}] (Direct products and sums in general)
\begin{lyxlist}{10.}
\item [{(a)}] Let $\left\{ V_{i}\right\} _{i\in I}$ be a family of vector
spaces, and let $\prod_{i\in I}V_{i}$ (their \emph{direct product})
denote the set $\left\{ f\colon I\to\bigcup_{i\in I}V_{i}\mid f(i)\in V_{i}\right\} $
(that is, the set of functions $f$ with domain $I$ such that $f(i)$
is an element of $V_{i}$ for all $i$). For $f,g\in\prod_{i\in I}V_{i}$
and $a,b\in\R$ define $af+bg$ by $\left(af+bg\right)(i)=af(i)+bg(i)$
(addition and multiplication in $V_{i}$). Show that this endows $\prod_{i\in I}V_{i\in I}$
with the structure of a vector space.
\item [{(b)}] Continuing with the same family, define the \emph{support}
of $f\in\prod_{i}V_{i}$ as $\supp(f)=\left\{ i\in I\mid f(i)\neq\zv_{V_{i}}\right\} $.
Show that the \emph{direct sum} $\bigoplus_{i\in I}V_{i}\eqdef\left\{ f\in\prod_{i}V_{i}\mid\supp(f)\textrm{ is finite}\right\} $
is a subspace (compare with problem 6(c)).
\item [{(c)}] When all the $V_{i}$ are equal to a fixed space $V$ we
sometimes write $V^{I}$ for the direct product $\prod_{i\in I}V$,
and $V^{\oplus I}$ for the direct sum $\oplus_{i\in I}V$. Verify
that this agrees with the notation in 6(c). What is $V$ there? \bigskip{}
\end{lyxlist}
\end{lyxlist}
\begin{center}
\textbf{Supplementary problems: fields}
\par\end{center}
Notation: $\forall$ means ``For all'' and $\exists$ mean ``there
exists''.
\begin{defn*}
A \emph{field} is a triple $\left(F,+,\cdot\right)$ of a set $F$
and two binary operations on $F$ so that there are elements $0,1\in F$
for which:
\[
\forall x,y,z\in F\,:\,x+y=y+x,\,(x+y)+z=x+(y+z),\,x+0=x,\,\exists x':x+x'=0
\]
\[
\forall x,y,z\in F\,:\,x\cdot y=y\cdot x,\,(x\cdot y)\cdot z=x\cdot(y\cdot z),\,x\cdot1=x,\,(x\neq0)\implies\exists\tilde{x}:x\cdot\tilde{x}=1
\]
\[
\forall x,y,z\in F:x\cdot(y+z)=x\cdot y+x\cdot z
\]
\end{defn*}
\begin{lyxlist}{10.}
\item [{C.}] (Elementary calculations) Let $F$ be a field.
\begin{lyxlist}{10.}
\item [{(a)}] Let $0_{1},0_{2}$ be two elements of $F$ which can be used
in the definition above. By considering the sum $0_{1}+0_{2}$ show
that $0_{1}=0_{2}$.
\item [{(b)}] Let $x\in F$ and let $x'_{1},x'_{2}\in F$ be such that
$x+x'_{1}=x+x'_{2}=0$. Adding $x_{1}'$ to both sides conclude that
$x'_{1}=x'_{2}$. This element is usually denoted $-x$.
\item [{(c)}] Let $x\in F$. Show that $0\cdot x=0$.
\item [{(d)}] Similarly show that $1$ and $\tilde{x}$ (usually denoted
$x^{-1}$) are unique.
\item [{(e)}] Show that if $xy=0$ then $x=0$ or $y=0$.\bigskip{}
\end{lyxlist}
\item [{D.}] Consider the set $\left\{ 0,1\right\} $ with $0\neq1$. Define
$1+1=0$, and define all other sums and products in this set as required
by the definition above or by C(c). Show that the result is a field.
Show that defining $1+1=1$ would not result in a field, and conclude
that there is a unique field with two elements, denoted $\FF_{2}$
from now on.
\end{lyxlist}
\begin{defn*}
A vector space over the field $F$ has the same definition as given
in class, except that the field of scalars $\R$ is replaced with
$F$.
\end{defn*}
\newpage{}
\begin{lyxlist}{10.}
\item [{E.}] Let $X$ be a set. To a subset $A\subset X$ associate its
\emph{indicator function} $1_{A}(x)=\begin{cases}
1 & x\in A\\
0 & x\notin A
\end{cases}$. Show that the map $A\mapsto1_{A}$ gives a bijection between the
\emph{powerset} $\cP(X)=\left\{ A\mid A\subset X\right\} $ and the
vector space $\FF_{2}^{X}$. Show that under this identification addition
in $\FF_{2}^{X}$ maps to the operation \emph{of symmetric difference
}of sets, defined by $A\Delta B=\left\{ x\mid x\in A\cup B,\,x\notin A\cap B\right\} $
(that is, $A\Delta B$ is the set of elements of $X$ that are in
\emph{exactly one} of $A,B$ but not both).
\item [{F.}] Let $F$ be a field with finitely many elements. For an integer
$n\geq0$ write $\bar{n}=\sum_{i=1}^{n}1_{F}$.
\begin{lyxlist}{10.}
\item [{(a)}] Show that $\bar{n}=\bar{m}$ for some $n>m>0$ and conclude
that $\bar{p}=0_{F}$ for some positive integer $p$.
\item [{(b)}] Show that the smallest positive $p$ such that $\bar{p}=0_{F}$
is a prime number. This is called the \emph{characteristic} of $F$
and denoted $\chr(F)$.
\item [{({*}c)}] Show that $\left\{ \bar{i}\mid0\leq i<\chr(F)\right\} $
is a subfield of $F$, usually denoted the \emph{prime field} of $F$.
\item [{RMK}] We will later show that if $F$ has characteristic $p$ then
its number of elements is of the form $q=p^{f}$ for some integer
$f$.
\end{lyxlist}
\end{lyxlist}
\end{document}