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\begin{center}\vspace{1in}{\bf MATH 209, MANIFOLDS II, WINTER 2011}\\
\bigskip
{\bf Homework Assignment III: Linear Algebra}
\end{center}
\bigskip
 
\noindent{\bf 1.}  [HAND IN] 
Let $\alpha_1,\ldots,\alpha_n$ be elements of $V^*$.
Define the matrix $a_{ij}$ by
$
\alpha_i=\sum_{j}a_{ij}e^*_j
$.
Prove that 
$\alpha_1\wedge\cdots\wedge\alpha_n=
\det(a_{ij})\,e^*_1\wedge\cdots\wedge e^*_n$.
\bigskip

\noindent{\bf 2.} 
Let $F\colon V\to V$ be a linear map. Prove that the map of
one-dimensional vector spaces
$F^*\colon \bigwedge^n V^* \to \bigwedge^n V^*$ is multiplication
by $\det F$.

\bigskip
 
\noindent{\bf 3.}  An element $\omega\in A^2(V)=\bigwedge^2 V^*$ is
called \emph{non-degenerate} or a \emph{linear symplectic form} on $V$
if $i_v\omega\neq 0$ for any non-zero $v\in V$.  Define the matrix
$A=(a_{ij})$ by $\omega=\sum_{i,j}a_{ij} \,e^*_i\otimes e^*_j $. Note
that $A$ is skew-symmetric, i.e., $A^\top=-A$, since $\omega$ is
skew-symmetric. 

\begin{enumerate}

\item[(a)]  Consider the map $V\to V^*$ given by $v\mapsto i_v \omega$.
Prove that this map is an isomorphism if and only if $\omega$ is 
non-degenerate.  Furthermore, $A$ the matrix of this map in the bases
$\{e_i\}$ and $\{e^*_i\}$. Thus, $\omega$ is non-degenerate if and only if
$\det A\neq 0$.

\item[(b)]  If $\omega$ is symplectic prove that  $dim V$ is even.

\item[(c)]  With $n = 2m = dim(V)$ as in (b), prove that $\omega$ is non-degenerate if and only if
$\omega^m\neq 0$ in $\bigwedge^n V^*$.

\item[(d)]  Let $\mu= e^*_1\wedge  e^*_2 \wedge \ldots  \wedge e^* _n$ be the standard volume form, 
and write $\omega^m = P \mu$ with $P \in \R$ and $2m = dim(V)$ as in item (c).   Show that $P^2 = det(A)$. 

{ \bf Remark.} The function 
$P = P(A)$ of (d)  is called the ``Pfaffian'', usually written $Pf(A)$.   It  is a polynomial in the entries of $A$
and defines an analytic square root for the determinant of  skew-symmetric matrices.  It appears in the Chern-Gauss-Bonnet formula. 
 

\end{enumerate}

\bigskip

\bigskip

\noindent{\bf 4.} Consider the map $\Phi\colon V\otimes W^* \to L(W,V)$ sending
$v\otimes \alpha$, where $v\in V$ and $\alpha \in W^*$, to the linear
map $W\ni x\mapsto \alpha(x)v\in V$. Prove that $\Phi$, define a  linear isomorphism. 

 

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