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\title{Math 728\\Fall 2004\\Homework 2}
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\centerline{\it Due Wednesday September 8, 2004.}

\centerline{}

\noindent 1. Let $W\subseteq V$ be vector spaces over $\mathbb{F}$.
\begin{itemize}
\item[a)] (5 pt) Show that $\text{dim}(W)\leq\text{dim}(V)$.
\item[b)] (5 pt) Show that $\text{dim}(V)=\text{dim}(W)+\text{dim}(V/W)$. 
\end{itemize}

\centerline{}

\noindent 2. (5 pt) Let $V$ and $W$ be subspaces of some vector space $U$. Show that 

\[
\text{dim}(V)+\text{dim}(W)=\text{dim}(V\bigcap W)+\text{dim}(V+W).
\]

\noindent What happens in the case where $U=V\oplus W$?


\centerline{}

\noindent 3. {\it In this problem we explore the notion of a ``dual space" which will reappear in many forms}. 

Let $V$ be a vector space over $\mathbb{F}$. We define $V^*=\text{Hom}_{\mathbb{F}}(V,\mathbb{F})$
\begin{itemize}
\item[a)] (5 pt) Show that $V^*$ is a vector space over $\mathbb{F}$.
\item[b)] (5 pt) Show that if $V$ is finite dimensional, then $V^*\cong V$.
\item[c)] (5 pt) Show that if $V$ is finite dimensional then there is an isomorphism $V\cong V^{**}$ that is independent of the choice of basis of $V$.
\item[d)] (5 pt) Show that if $V$ is infinite dimensional then $V^{*}\ncong V$.
\end{itemize}



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