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\newcommand{\myTitle}{HW 3, due Friday, February 22}
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\begin{enumerate}
\item
  \begin{enumerate}
    \item Let $X$ and $Y$ be sets.  What are $X\times Y$ and $X\oplus Y$?
      Show they satisfy the appropriate universal properties.
    \item Let $X$ and $Y$ be objects in any category.  If $A$ and $B$ are both
      products of $X$ and~$Y$, use the universal property of products to show
      that $A$ and $B$ are isomorphic. 
  \end{enumerate}
  {\sc solution:}

\begin{center}
  \begin{tikzpicture}[node distance=2cm, auto, >=latex']
  \node (Z) {$Z$};
  \node (XxY) [below of=Z] {$X\times Y$};
  \node (X) [below left of=XxY] {$X$};
  \node (Y) [below right of=XxY] {$Y$};
  \draw[->] (XxY) to node [swap] {$\pi_X$} (X);
  \draw[->] (XxY) to node {$\pi_Y$} (Y);
  \draw[->, bend right=40] (Z) to node [swap] {$f$} (X);
  \draw[->, bend left=40] (Z) to node {$g$} (Y);
  \draw[densely dotted,->,thick] (Z) to node {$\exists!h$} (XxY);
\end{tikzpicture}
\end{center}
\item Let $V$ be a vector space over a field $K$, and let
  $v_1,\dots,v_n$ be a basis for $V$.  For $i=1,\dots,n$, define
  mappings $v_i^*\colon V\to K$ by requiring
  $v_i^*(v_j)=\delta(i,j)$ and extending linearly.  Show that
  $v_1^*,\dots,v_n^*$ is a basis for $V^*=\mbox{hom}(V,K)$.
  
  {\sc solution:}

\item Let $L\colon\R^n\to\R^m$ be a linear function.  The function $L$
  is represented by the matrix $A$ whose $j$-th column is $L(e_j)$,
  the image of the $j$-th standard basis vector.
  Choosing the basis dual to the standard basis, show that the matrix
  representing $L^*\colon (\R^m)^*\to (\R^n)^*$ is $A^t$, the transpose of $A$.

  {\sc solution:}


\item The $n$-sphere is the set $S^n=\{x\in\R^{n+1}\mid |x|=1\}$. It
  has a topology induced by $\R^{n+1}$, i.e., a set is open in $S^n$
  iff it is the intersection of an open set of $\R^{n+1}$ with $S^n$.
  Each point in $S^n$ has some non-zero coordinate.  For $i=1,\dots,n+1$, define
  $U_i^{+}=\{x\in\ S^n:x_i>0\}$ and $U_i^{-}=\{x\in\ S^n:x_i<0\}$.
  Define $\pi_i(x_1,\dots,x_{n+1})=(x_1,\dots,x_{i-1},x_{i+1},\dots,x_{n+1})$.
  Then the collection of charts $(U_i^+,\pi_i)$ and $(U_i^-,\pi_i)$ for
  $i=1,\dots,n+1$ serves as an atlas for $S^n$.
\begin{enumerate}
\item Show that the charts constructed in this way are differentially
  compatible.  
\item Consider the function $f(x)=\sum_{i=1}^{n+1}x_i^4$ defined on
  $S^n$.  Let $p\in S^n$ with $p_1>0$.  With respect to the chart
  $(U_1^+,\pi_1)$, take the local coordinates to be $x_2,\dots,x_{n+1}$, as
  seems natural in this case.  Let
\[
v:=a\left(\frac{\partial}{\partial x_2}\right)_p + b\left(\frac{\partial}{\partial x_3}\right)_p
\]
be a tangent vector at $p$.  Calculate $v(f)$.  In other words, think of $v$
as a derivation and apply it to $f$.
\end{enumerate}

  {\sc solution:}

\item Let $V=W=\R^3$ and let $L\colon V\to W$ be a linear mapping with
  matrix $A=(a_{ij})$ relative to the standard bases bases $v_1,v_2,v_3$ for
  $V$ and $w_1,w_2,w_3$ for $W$.  Of course, $v_i=w_i=e_i$ for all $i$,
  but it is useful to have separate notation.  Let
\[
\omega=w_1^*\wedge w_3^*\in\Lambda^2W^*.
\]
\begin{enumerate}
\item Corresponding to $\omega$ is a bilinear, alternating form
  $\tilde{\omega}\in\mbox{Alt}^2W$:
\[
\tilde{\omega}\colon W\times W\to\R.
\]
Describe $\tilde{\omega}$ by calculating the images of $(w_i,w_j)$ for
$1\leq i<j\leq 3$.
\item Find $\alpha,\beta,\gamma$ as functions of the entries of $A$
  so that $L^*\omega=\alpha\,v_1^*\wedge v_2^*+\beta\,v_1^*\wedge
  v_3^*+\gamma\,v_2^*\wedge v_3^*$. Do this by computing $(L^*\omega)(v_i\wedge
  v_j)$ for each $i<j$.
\item The mapping $L$ is given by 
\[
L(x,y,z)=
(a_{11}x+a_{12}y+a_{13}z,
a_{21}x+a_{22}y+a_{23}z,
a_{31}x+a_{32}y+a_{33}z).
\]
Calculate
\[
(a_{11}v_1^*+a_{12}v_2^*+a_{13}v_3^*)\wedge
(a_{31}v_1^*+a_{32}v_2^*+a_{33}v_3^*)
\]
and compare with your answer to the previous problem. 
\end{enumerate}
  {\sc solution:}

\end{enumerate}
\end{document}
