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\title{Math Studies Homework 2}
\author{James Cummings}

\begin{document}

\maketitle


    Submit the \LaTeX\ file containing your solutions to the
    email address algebrahw@legba.math.cmu.edu by
    1159 pm on Fri 4 Feb.
 


\begin{enumerate}


\item  Let $U$, $V$ and $W$ be VS's (not necessarily finite dimensional) over a field $F$. Show that
 the spaces  $Hom(U, Hom(V, W))$ and $Hom(U \otimes V, W)$ are isomorphic
 {\em without} choosing bases.


\item  Let $V$ be a VS over $F$ of dimension $m$.   Show that
       if $v_1, \ldots v_m \in V$ then the $v_i$ are
       independent iff $v_1 \wedge \ldots v_m \neq 0$ in the
       exterior power $\bigwedge^m V$. 
       Hint:   you may wish to choose a basis $w_1, \ldots w_m$
       and consider the unique linear map $T$ taking $w_i$ to
       $v_i$.



\item  Let $E$ and $F$ be fields with $E$ a subfield of $F$, and let
      $V$ be a VS over $E$. 

\begin{enumerate}

\item  Show that $F$ is a VS over $E$ if we define $+$ on $F$ to be the field addition in $F$,
       and define scalar multiplication by the restriction of field multiplication in $F$
       to $E \times F$.   


\item  Let $W = F \otimes_E V$, that is the $E$-VS obtained by forming the
    tensor product of the $E$-VS's  $F$ and $V$.  

\begin{enumerate}

\item  Show that if $\lambda \in F$ then the map from $F \times V$ to
       $W$ given by $(f, v) \mapsto (\lambda f) \otimes v$ is a bilinear
       map of $E$-VS's.  Deduce that there is a unique $E$-linear map
       $\psi_\lambda$ from $W$ to $W$ such that 
       $\psi_\lambda(f \otimes v) = (\lambda f) \otimes v$ for all
       $f$ and $v$.

\item  Show that if we define a scalar multiplication map from
       $F \times W$ to $W$ by $\lambda w = \psi_\lambda(w)$ 
       then $W$ becomes an $F$-VS.

\end{enumerate}

\item   Show that there is an $F$-subspace $X$ of $W$ which (when we regard it as
        a $E$-VS by restricting the scalar multiplication map to $E \times X$)
        is isomorphic to $V$.


\end{enumerate}


\item             Let $\mathcal C$ be a category and let $a$ and $b$ be objects of $\mathcal C$. Say that
                    a {\em diagram} over $\{ a, b \}$ is a pair $D$ of morphisms $(f, g)$ where 
                    $f: c \rightarrow a$ and $g: c \rightarrow b$ for some object $c$ of $\mathcal C$.


\begin{enumerate}

\item                    Show that we can construct a category $\mathcal D$ such that the objects of
                         $\mathcal D$ are the diagrams over $\{ a, b \}$, and the morphisms in $\mathcal D$ from the
                         diagram $(f, g)$ to the diagram $(f', g')$ are the  morphisms $h$ of $\mathcal C$ such that
                         $f' \circ h = f$ and $g' \circ h = g$. 


\item                   A {\em product} of $a$ and $b$ is a diagram $(f, g)$ over $\{ a, b \}$ such that 
                        for any diagram $(f', g')$ over $\{ a, b \}$ there is a unique morphism  $h$ of $\mathcal C$ 
                        such that $f' = f \circ h$ and $g' = g \circ h$. 
                        Show that if $(f_1, g_1)$ and $(f_2, g_2)$
                        are both products of $\{ a, b \}$ there is a unique isomorphism 
                        $h$ such that $f_2 \circ h = f_1$ and $g_2 \circ h = g_1$. 

\item                   Show that products always exist in the category of groups and  group HMs, and the
                        category of sets and functions. Find a category in which not all products exist. 


\end{enumerate}


\item   Let $V$ be a space of dimension $m$ and let $n \le m$. Show that the exterior power
        $\bigwedge^n V$ has dimension $m \choose n$. 



\end{enumerate}



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