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\begin{enumerate}
\item Let $V$ and $W$ be vector spaces over an arbitrary field $K$,
  and suppose that $V$ has finite dimension $n$.
\begin{enumerate}
\item Show that 
\[
\mbox{hom}(V,W)\approx W^n=\underbrace{W\times\dots\times
  W}_{n\mathrm{\ factors}}.
\]
\item There is a mapping of vector spaces
  \begin{align*}
V^*\otimes W&\to\mbox{hom}(V,W)\\
\phi\otimes w&\mapsto[ v\mapsto \phi(v)w]
\end{align*}
It's induced by the corresponding bilinear mapping from $V^*\times
W$ via the universal property for tensor products.  Show that this mapping is an isomorphism.  [We use
  finite-dimensionality here.  It is useful to pick a basis for $V$
  and its corresponding dual basis.] 
\item The cross product 
  \begin{align*}
\R^3\times\R^3&\to\R^3\\
(u,v)&\mapsto u\times v
\end{align*}
is an alternating, multilinear mapping.  Hence, it factors through a
unique mapping $\Lambda^2\R^3\to\R^3$, i.e., to an element of
$\mbox{hom}(\Lambda^2\R^3,\R^3)$.  By the previous exercise, we can
identify this element with an element of
$(\Lambda^2\R^3)^*\otimes\R^3$, and hence with an element of
$\Lambda^2((\R^3)^*)\otimes\R^3$.  Letting $e_1,e_2,e_3$ be the standard
basis of $\R^3$, the space $\Lambda^2((\R^3)^*)\otimes\R^3$ has a basis
$\{(e_i^*\wedge e_j^*)\otimes e_k\}$ where $1\leq i<j\leq 3$ and $1\leq
k\leq 3$.  Identify the cross product in terms of this basis.
\end{enumerate}
\item Orientation.
  \begin{enumerate}
  \item Is the standard atlas for $\proj^1$ an orienting atlas?  If not, can you
    fix it?
  \item Same question for $\proj^3$.
  \end{enumerate}
\item Consider the differential form
  \[
\eta = x\,dy\wedge dz-y\,dx\wedge dz+z\,dx\wedge dy\in\Omega^2\R^3.
  \]
  Let $\iota\colon S^2\to\R^3$ be the standard embedding of the $2$-sphere, and
  let $\omega=\iota^*\eta\in\Omega^2S^2$.  Compute $\int_{S^2}\omega$ by
  choosing charts and using the definition of integration.  Check your result
  with Stokes' theorem.
\item Let $X$ be a topological space, and let $Y$ be a subset of $X$.
  The {\em subspace topology} on~$Y$ is given by the collection of
  sets $\{Y\cap U\colon\mbox{$U$ open in $X$}\}$.  Thus, a set in $Y$
  is open in the subspace topology if and only if it is the
  intersection of an open subset of $X$ with~$Y$.  The term ``open
  set'' can be ambiguous in this context; to be clear, you can say
  ``open in $Y$'' or ``open in $X$''.

Let $Y=[-1,1]$ as a subspace of $\R$.  Which of the following
  are open in $Y$, and which are open in $\R$?
\begin{enumerate}
\item $A=\{x\colon \frac{1}{2}<|x|<1\}$.
\item $B=\{x\colon \frac{1}{2}<|x|\leq 1\}$.
\item $C=\{x\colon \frac{1}{2}\leq|x|<1\}$.
\item $D=\{x\colon \frac{1}{2}\leq|x|\leq1\}$.
\item $E=\{x\colon 0<|x|<1 \mbox{ and $\frac{1}{x}\notin\Z_{>0}$}\}$.
\end{enumerate}
\item Let $X$ and $Y$ be topological spaces.  The {\em product
  topology} on $X\times Y$ is the topology with basis $U\times V$
  where $U$ and $V$ are open sets in $X$ and $Y$, respectively.
Is it true that every open set in the product topology has the form
  $U\times V$ with $U$ and $V$ open in $X$ and $Y$, respectively?
  Proof or counterexample.
\item A function $f\colon X\to Y$ between topological spaces is {\em
  continuous} if $f^{-1}(V)$ is open in $X$ for each open set $V$ in $Y$.
  The function $f$ is {\em open} of $f(U)$ is open in $Y$ for each open set
  $U$ in $X$.  
\begin{enumerate}
\item Let $X$ and $Y$ be topological spaces, and consider $X\times Y$
  with the product topology.  Let $\pi\colon X\times Y\to X$ be the
  first projection: $\pi(x,y)=x$.  
\begin{enumerate}
\item Show that $\pi$ is continuous.
\item Show that $\pi$ is open.
\end{enumerate}
\item (Note: the following exercises would work just as well if one substituted
  the word ``homeomorphism'' for ``diffeomorphism'' everywhere.  However, the
  result arose in the context of diffeomorphisms in class.) Let $X\subset\R^n$ and $Y\subset\R^m$ be open sets, and let
  $f\colon X\to Y$ be a local diffeomorphism.  This means that for
  each $x\in X$ there is an open neighborhood $U_x\subset X$ of $x$
  such that $f(U_x)$ is open and $f|_{U_x}$ is a diffeomorphism onto $f(U_x)$.  
  \begin{enumerate}
\item Give a simple example of a local diffeomorphism which is not a
  diffeomorphism.
\item Show that a local diffeomorphism is an open mapping.
\end{enumerate}
\end{enumerate}
\item Let $V$ and $W$ be vector spaces over a field $k$, and suppose $L\colon
  V\to W$ is a linear function.  We have defined
\begin{eqnarray*}
  L^*\colon \Lambda^{\ell}W^*&\to&\Lambda^{\ell}V^*\\
  \phi_1\wedge\dots\wedge\phi_{\ell}&\mapsto& (\phi_1\circ
  L)\wedge\dots\wedge(\phi_{\ell}\circ L)
\end{eqnarray*}
Show that for all $v_1,\dots,v_{\ell}\in V$,
\[
L^*(\phi_1\wedge\dots\phi_{\ell})(v_1,\dots,v_{\ell})=
(\phi_1\wedge\dots\wedge\phi_{\ell})(Lv_1,\dots,Lv_{\ell}).
\]
\end{enumerate}
\end{document}
