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\begin{document}

\title{Differential Topology Problem Set \#4}
%
\author{Due: Tuesday, April 5}
\date{}
\maketitle

\begin{enumerate}
\item Consider the map
$$
V\times W \rightarrow V\oplus W, \qquad v\times w\to v\oplus w.
$$ Is this map bilinear?

\item Suppose $v_1,\dots, v_p\in V$ are linearly dependent vectors. Show that 
$$
T(v_1,\dots, v_p)=0
$$
for all $T\in \Lambda^p(V^*)$. Is this true for all $T\in \mathcal{J}^p(V^*)$? If so, prove it, and if not, find a counterexample.
\item Chapter 10, Section 2, \#3.

\item Chapter 10, Section 2, \#10 (a) and (b).

\item Chapter 10, Section 3, Exercise on p. 165.

\newcounter{number}\setcounter{number}{\value{enumi}}
\end{enumerate}
{\bf The Tensor Product of Vector Spaces.} Let $V$ and $W$ be vector spaces over a field (you may assume they are real vector spaces). The tensor product
$V\otimes W$ is a vector space equipped with a bilinear map
$$
V\times W \longrightarrow V\otimes W,\qquad v\times w \to v\otimes w
$$
(for every $v\in V, w\in W$) which is {\em universal} in the following sense. For any bilinear map
$$
b: V\times W \longrightarrow U
$$
where $U$ is a vector space (over the same field), there is a unique linear map
$$
L: V\otimes W\longrightarrow U
$$
such that $L(v\otimes w) = b(v,w)$. In other words, any linear map $b: V\times W\longrightarrow U$ factors through the tensor product $V\otimes W$. Another way of saying this is that the diagram
$$
\begin{array}{c}
\xymatrix{ V\times W \ar[r] \ar@{->}[rd]_b& V\otimes W \ar@{->}[d]_{L}\\
& U
}
\end{array}
$$
commutes for every vector space $U$ and for every bilinear map $b$.

\begin{enumerate}
\setcounter{enumi}{\thenumber}
\item Find a basis of $V\otimes W$ (given a basis of $V$ and of $W$). Find the dimension of $V\otimes W$.
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\end{enumerate}



\newpage

\begin{center}
{\bf Additional problems for graduate students, or undergraduate extra credit}
\end{center}
The {\em $n$th exterior power} of a vector space $V$ is a vector space $Alt^nV$,  equipped with an alternating multilinear map
$$
V\times\cdots\times V\rightarrow Alt^nV,\qquad v_1\times\cdots\times v_n \to v_1\odot\cdots \odot v_n,
$$ 
that is universal in the following sense. For any alternating multilinear map $b: V\times\cdots\times V\rightarrow U$ (where $U$ is a vector space), there is a unique linear map $L: Alt^nV\rightarrow U$ which takes $v_1\odot\cdots \odot v_n$ to $b(v_1,\dots, v_n)$.
\begin{enumerate}
\setcounter{enumi}{\thenumber}

\item Show that $\Lambda^n(V^*) \iso Alt^nV^*$, where $V^*$ is the dual to $V$ (and $\Lambda^n(V^*)$ is the same as we defined in class). {\em Hint.} Show that $\Lambda^n(V^*)$ has the universality property, or use the universality property of $Alt^nV^*$ to construct a map between the two spaces and then prove it's an isomorphism.
\item Show that $\Lambda^n(V)$ can be constructed as a quotient of $V\otimes V\otimes\cdots \otimes V$ ($n$ times). In other words, there is a surjective map
$$
L: V^{\otimes n}\longrightarrow \Lambda^n(V).
$$
Write down the map, show it's surjective, and find its kernel.
\end{enumerate}
{\bf NOTE:} The product $\odot$ is written $\wedge$ because it is in fact the same product as the one we use in class!

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