\documentclass[10pt]{article}

% Identical to 2010 and 2011

\usepackage[margin=.8in,nohead]{geometry}


%\topmargin -1in \textwidth 7in \textheight 9.8in \oddsidemargin
%-.25in \pagestyle{empty}
%\parindent 0in

\usepackage{amsmath,amsfonts}
\usepackage{amsthm}


\theoremstyle{definition}
\newtheorem{definition}{Definition}%[section]
\newtheorem{problem}[definition]{Problem}
\newtheorem{example}[definition]{Example}
\newtheorem{exercise}[definition]{Exercise}
\newtheorem{remark}[definition]{Remark}

\theoremstyle{plain}
\newtheorem{lemma}[definition]{Lemma}
\newtheorem{theorem}[definition]{Theorem}
\newtheorem{proposition}[definition]{Proposition}
\newtheorem{corollary}[definition]{Corollary}

\newcommand{\nA}{\mathbb A}
\newcommand{\nB}{\mathbb B}
\newcommand{\nC}{\mathbb C}
\newcommand{\nD}{\mathbb D}
\newcommand{\nE}{\mathbb E}
\newcommand{\nF}{\mathbb F}
\newcommand{\nG}{\mathbb G}
\newcommand{\nH}{\mathbb H}
\newcommand{\nI}{\mathbb I}
\newcommand{\nJ}{\mathbb J}
\newcommand{\nK}{\mathbb K}
\newcommand{\nL}{\mathbb L}
\newcommand{\nM}{\mathbb M}
\newcommand{\nN}{\mathbb N}
\newcommand{\nO}{\mathbb O}
\newcommand{\nP}{\mathbb P}
\newcommand{\nQ}{\mathbb Q}
\newcommand{\nR}{\mathbb R}
\newcommand{\nS}{\mathbb S}
\newcommand{\nT}{\mathbb T}
\newcommand{\nZ}{\mathbb Z}

\newcommand{\cA}{{\mathcal A}}
\newcommand{\cB}{{\mathcal B}}
\newcommand{\cC}{{\mathcal C}}
\newcommand{\cD}{{\mathcal D}}
\newcommand{\cE}{{\mathcal E}}
\newcommand{\cF}{{\mathcal F}}
\newcommand{\cG}{{\mathcal G}}
\newcommand{\cH}{{\mathcal H}}
\newcommand{\cI}{{\mathcal I}}
\newcommand{\cK}{{\mathcal K}}
\newcommand{\cL}{{\mathcal L}}
\newcommand{\cM}{{\mathcal M}}
\newcommand{\cN}{{\mathcal N}}
\newcommand{\cO}{{\mathcal O}}
\newcommand{\cP}{{\mathcal P}}
\newcommand{\cQ}{{\mathcal Q}}
\newcommand{\cR}{{\mathcal R}}
\newcommand{\cS}{{\mathcal S}}
\newcommand{\cT}{{\mathcal T}}
\newcommand{\cU}{{\mathcal U}}
\newcommand{\cV}{{\mathcal V}}
\newcommand{\cW}{{\mathcal W}}
\newcommand{\cX}{{\mathcal X}}
\newcommand{\cY}{{\mathcal Y}}
\newcommand{\cZ}{{\mathcal Z}}

\newcommand{\cnj}[1]{\overline{#1}}
\newcommand{\V}[1]{\mathbf{#1}}
\newcommand{\M}[1]{\mathsf{#1}}
\newcommand{\R}[1]{\mathrm{#1}}

\DeclareMathOperator{\lspan}{span} \DeclareMathOperator{\ospan}{span}
\DeclareMathOperator{\cspan}{\overline{span}}
\DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\ran}{ran}
\DeclareMathOperator{\dom}{dom} \DeclareMathOperator{\fdom}{fdom}
\DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\Ker}{ker}
\DeclareMathOperator{\re}{Re} \DeclareMathOperator{\im}{Im}
\DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\nul}{nul}
\DeclareMathOperator{\lpd}{lpd}


\pagestyle{empty}
\parindent 0in


\begin{document}

\newcommand{\head}[2]{Math 504 \hfill Assignment 1
              \hfill October 15, 2013
                   \hfill Page #1 of #2 \\[.2in]}
\newcounter{hash}

\parbox[b]{1.6in}{\Huge MATH 504} \
\parbox[b]{2in}{\normalsize {Assignment 1   \\
October 15, 2013}} \ \hfill Name \rule{2.25in}{.5pt}

 \normalsize
%

\begin{problem}
Let  $D$ be a finite set and let $\nF$ be a scalar field. Then the set of all functions defined on $D$ with values in $\nF$ is a vector space over $\nF$ with the addition and scalar multiplication of functions defined pointwise. This space is denoted by $\nF^D$.
\begin{enumerate}
\renewcommand*\theenumi{\alph{enumi}}
\renewcommand*\labelenumi{(\theenumi)}
\item
Prove that $\nF^D$ is finite dimensional if and only if $D$ is finite. \item
If $D$ is finite, then $\dim \bigl(\nF^D\bigr) = |D|$.
\end{enumerate}
\end{problem}


\begin{problem} %3
Consider the vector space $\nR^\nR$ of all real valued
functions defined on $\nR$. This vector space is considered over the field $\nR$.  The purpose of this exercise is to study some special subspaces of the vector space $\nR^\nR$. Let $\gamma$ be an arbitrary (fixed) real number. Consider the set
\[
\cS_\gamma := \Big\{ f \in \nR^\nR: \exists \ a, b \in \nR \ \ \text{such that} \ \  f(t) = a \sin(\gamma t+b) \ \ \forall \  t\in\nR \Big\} .
\]
\begin{enumerate}
\renewcommand*\theenumi{\alph{enumi}}
\renewcommand*\labelenumi{(\theenumi)}
\item
Do you see exceptional values for $\gamma$ for which the set $\cS_\gamma$ is particularly simple?
\item
Prove that $\cS_\gamma$ is a subspace of $\nR^\nR$.
\item
For each $\gamma \in \nR$ find a basis for $\cS_\gamma$.  Plot the function $\gamma \mapsto \dim \cS_\gamma$.
\end{enumerate}
\end{problem}

\begin{problem}
Let $\cV$ be a vector space over $\nF$. Let $\cA$ be a linearly independent subset of $\cV$. Let $u \in \cV$ be arbitrary. By $u + \cA$ we denote the set of vectors $\{u+v : v \in \cA \}$.
\begin{enumerate}
\renewcommand*\theenumi{\alph{enumi}}
\renewcommand*\labelenumi{(\theenumi)}
\item
Prove the following implication. If $w \notin \lspan \cA$, then $ w + \cA$ is a linearly independent set.
\item
Is the converse of the implication in (a) true?
\item
Let $\alpha_1,\cdots,\alpha_n \in \nF$, let $v_1,\ldots, v_n$ be distinct vectors in $\cA$ and let $w = \alpha_1 v_1 + \cdots + \alpha_n v_n$. Find a necessary and sufficient condition (in terms of $\alpha_1, \ldots, \alpha_n$) for the linear independence of the
vectors $v_1 + w, \ldots, v_n + w$.
\end{enumerate}
\end{problem}

\begin{problem}
Let $\cV$ and $\cW$ be vector spaces over a scalar field $\nF$.
Assume that $\dim \cV = m$ and $\dim \cW = n.$  Let
$T:\cV\rightarrow\cW$ be a linear mapping. Prove that we can choose
a basis $\cB$ of $\cV$ and a basis $\cC$ of $\cW$ such that for some
integer $k, \ 0\leq k \leq \min\{m,n\}$ we have
 \begin{equation*} %\label{rref}
 \M{M}^{\cB}_{\cC}(T)=\left[ \begin{array}{ll}
 \M{I}_{k\times k} & \M{0}_{k \times (m-k)} \\
  \M{0}_{(n-k)\times k} &  \M{0}_{(n-k)\times (m-k)}
\end{array} \right] \ .
 \end{equation*}
\end{problem}

\begin{problem}
Let $\cV$ be a finite dimensional vector space over a field $\nF$
and let $T: \cV \rightarrow \cV$ be a linear mapping.  Put $T^{2} :=
T \circ T$.
\begin{enumerate}
\renewcommand*\theenumi{\alph{enumi}}
\renewcommand*\labelenumi{(\theenumi)}
\item
Discover and prove an inclusion relation between $\nul(T)$ and
$\nul(T^2)$, and between $\ran(T)$ and $\ran(T^2)$.
\item
If $\ran(T)=\ran(T^2)$, prove that $(\ran(T))\cap (\nul(T))
= \{0_\cV\}$.
\item
If $\ran(T)=\ran(T^2)$, prove that $\cV$ is a direct sum of $\ran(T)$ and $\nul(T)$.
\end{enumerate}
\end{problem}

\begin{problem}
Let $\cV$ be a vector space over $\nF$ and $T \in \cL(\cV)$.
Assume that that there exists a function $f: \cV \rightarrow \nF$
such that $Tv = f(v)\, v$ for each $v \in \cV$.  Prove that $T$ is a
multiple of the identity mapping, that is, there exists $\alpha \in
\nF$ such that $Tv = \alpha\, v$ for each $v \in \cV$. (A plain English explanation: The equation $Tv = f(v)\, v$ is telling us that $T$ scales each vector in $\cV$ by the scaling coefficient $f(v)$. The point of the problem is to prove that $T$ must scale each vector by the same coefficient. This is a consequence of the linearity of $T$.)
\end{problem}


\begin{problem}
Let $\cU, \cV$ and $\cW$ be nontrivial finite dimensional vector spaces over a scalar field $\nF$.
\begin{enumerate}
\renewcommand*\theenumi{\alph{enumi}}
\renewcommand*\labelenumi{(\theenumi)}
\item \label{isuperS} 
Let $S \in \cL(\cU,\cV)$ be an arbitrary fixed operator. Define
\[
{\mathbf S}: \cL(\cV,\cW) \to \cL(\cU,\cW) \qquad \text{by} \qquad {\mathbf S}(T) = TS, \quad T \in \cL(\cV,\cW).
\]
Prove that ${\mathbf S}$ is injective iff $S$ is surjective. Prove that ${\mathbf S}$ is surjective iff $S$ is injective.
\item
Let $T \in \cL(\cV,\cW)$ be an arbitrary fixed operator. Define
\[
{\mathbf T}: \cL(\cU,\cV) \to \cL(\cU,\cW) \qquad \text{by} \qquad {\mathbf T}(S) = TS, \quad S \in \cL(\cU,\cV).
\]
In (\ref{isuperS}) we characterized injectivity and surjectivity of  ${\mathbf S}$.  Formulate and prove an analogous statement for ${\mathbf T}$.
\end{enumerate}
\end{problem}

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{problem}[Hard]  %
Let a vector space $\cV$ of dimension $n$ be a direct sum of
vector spaces $\cU$ and $\cW$ of dimensions $k$ and $m$,
respectively.  Let $\{ v_1 , \ldots , v_n \}$ be a basis of $\cV$
such that $v_j = u_j + w_j $ where $u_j \in \cU$ and $w_j \in \cW,
$ with $j = 1, \ldots , n$. Then, after a suitable renumbering of
$v_1, \ldots , v_n, $  the set $\{u_1, \ldots , u_k \}$ is a basis
of $\cU$ and $\{ w_{k+1}, \ldots , w_n \}$ is a basis of $\cW$.
\end{problem}


\begin{problem}%2
Consider the vector space $\cP_2 = \cP_2(\nR)$  (over the field of
real numbers $\nR$) of all polynomials with real coefficients of
degree smaller or equal than 2, defined on the real line. We say
that $p \in \cP_2$ has a {\em vertex at} $t \in \nR$ if $p(t) \leq
p(x)$ for all $x \in \nR$ or $p(t) \geq p(x)$ for all $x \in \nR$.

\begin{enumerate}[(a)]
\item
Let $s \in \nR$ be an arbitrary number. Let $\cZ_{s}$ be the set of
all polynomials $p \in \cP_2$ such that $p(s) = 0$, that is,
\[
\cZ_{s} :=\{ p \in\cP_2 : p(s) = 0 \}.
\]
Prove that $\cZ_{s}$ is a subspace of $\cP_2$. Find a basis of this
subspace.  What is $\dim \cZ_{s}$?
\item %
Let $t \in \nR$. Let $\cV_{t}$ be the set of all polynomials $p \in
\cP_2$ which have a vertex at $t$.  Prove that $\cV_{t}$ is a
subspace of $\cP_2$. Find a basis of this subspace.  What is $\dim
\cV_{t}$?
\item
Let $s, t \in \nR$, $s \neq t$.  Describe the polynomials in each of
the subspaces $\cZ_{s} \cap \cZ_t$, $\cV_t \cap \cZ_s$ and $\cV_s
\cap \cV_t$. Find a basis for each of these subspaces.
\item
Let $s, t \in \nR$ be given. Solve the equation $\cZ_{s} \cap \cZ_x
= \cV_y \cap \cZ_{t}$ for $x$ and $y$.

\end{enumerate}
\end{problem}
