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%%%%%% Math 403 Homework \#5, Spring 2024 %%%%%%
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%%%%%% Instructor: Ezra Miller %%%%%%
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\title{\mbox{}\\[-8ex]Math 403 Homework \#5, Spring 2024\\\normalsize
Instructor: Ezra Miller\\[-2.5ex]}
\author{Solutions by: ...your name...\\[1ex]
Collaborators: ...list those with whom you worked on this assignment...\\
(1 point for each of up to 3 collaborators who also list you)}
\date{Due: 11:59pm Tuesday 9 April 2024}
\maketitle
\vspace{-5ex}\collab{}%
\noindent
\textsc{Reading assignments}\total{}{36}%covers Lectures 20 - 22
\begin{enumerate}[itemsep=-1ex]\setcounter{enumi}{19}
\item%20.
for Tue.~26 March
\begin{itemize}
\item{}%
[Treil, \S 8.5] multilinear algebra, tensor product
\item{}%
[Wikipedia, \emph{Tensor product}]
\end{itemize}
\item[21--22.]\addtocounter{enumi}{2}%21. and 22.
for Thu.~28 March and Tue.~2 April
\begin{itemize}
\item{}%
[Wikipedia, \emph{Exterior algebra}]
\end{itemize}
\end{enumerate}
\noindent
\textsc{Exercises}% covers Lectures 18 - 22
\begin{enumerate}
\item\ \score{}%1.
For $P \in \RR^{n \times n}$ with $P > 0$, set $\Gamma(P) = \{\lambda
\in \RR_{\geq 0} \mid P\xx \leq \lambda\xx \text{ for some nonzero }
\xx \geq \0\}$. Show that the dominant eigenvalue $\lambda(P)$
satisfies $\lambda(P) = \min_{\lambda\in\Gamma(P)} \lambda$.
\item\ \score{}%2.
In class, we stated the theorem that for $P > 0$ a stochastic $n
\times n$ matrix with dominant eigenvector~$\vv$, and $\xx \geq 0$ any
nonzero vector, $P^k\xx \to \alpha\vv$ as $k\to\infty$ for some real
$\alpha > 0$. But we only proved it when $P$ is diagonalizable.
Complete the proof.
\item\ \score{}%3.
Prove that $P$ has a dominant positive eigenvalue if $P \geq 0$ and
$P^k > 0$ for some $k > 0$.
\item\ \score{}%4.
Prove or disprove: the set of stochastic $n \times n$ matrices is
compact and convex.
\item\ \score{}%5.
Use the universal property of tensor products to prove commutativity:
there is a unique isomorphism $V \otimes W \to W \otimes V$ such that
$v \otimes w \mapsto w \otimes v$ for all $v \in V$ and~$w \in W$.
\item\ \score{}%6.
Use the universal property of tensor products to prove associativity:
there is a unique isomorphism $(U \otimes V) \otimes W \to U \otimes
(V \otimes W)$ such that $(u \otimes v) \otimes w \mapsto u \otimes (v
\otimes w)$ for all $u \in U$, $v \in V$, and $w \in W$. Hint: You
can either use the universal property to produce the map or check that
the two parenthesizations have the same universal property regarding
bilinear maps on $(U \times V) \times W = U \times (V \times W)$ and
appeal to ``abstract nonsense'': universal constructions are unique up
to unique isomorphism.
\item\ \score{}%7.
Prove that homomorphisms $\varphi: V \to V'$ and $\psi: W \to W'$
result in a canonical homomorphism $\varphi \otimes \psi: V \otimes W
\to V' \otimes W'$. Given matrices for $\varphi$ and $\psi$, write
down a matrix for $\varphi \otimes \psi$. Note: your answer will
depend on how you order the basis of~$V \otimes W$.
\item\ \score{}%8.
Prove that a homomorphism $\varphi: V \to W$ results in a canonical
homomorphism $\bigwedge^r \varphi: \bigwedge^r V \to \bigwedge^r W$.
Given a matrix for~$\varphi$, write down a matrix for $\bigwedge^r
\varphi$. Note: make no attempt to draw a matrix; just describe its
entries as labeled by pairs of basis vectors.
\item\ \score{}%9.
Prove that tensor products commute with direct sums: if $I$ is any
(finite or infinite) index set and $V = \bigoplus_{i \in I} V_i$,
then there is a natural isomorphism $V \otimes W \to\bigoplus_{i\in I}
V_i \otimes W$.
\item\ \score{}%10.
Construct a natural map $V^* \otimes W^* \to (V \otimes W)^*$. Show
that it is injective. If one of $V$ and $W$ has finite dimension,
show that the map is an isomorphism.
\item\ \score{}%11.
Prove the existence of a bilinear map $\bigwedge^r V \times
\bigwedge^s V \to \bigwedge^{r+s} V$ taking%
$$%
(v_1\wedge \cdots\wedge v_r, v'_1\wedge \cdots\wedge v'_s) \mapsto
v_1 \wedge \cdots\wedge v_r\wedge v'_1\wedge \cdots\wedge v'_s.
$$
Write $\omega = v_1\wedge \cdots\wedge v_r$ and $\omega' = v'_1\wedge
\cdots \wedge v'_r$, so $\omega \wedge \omega' = v_1 \wedge
\cdots\wedge v_r\wedge v'_1\wedge \cdots\wedge v'_s$. Show that
$\omega' \wedge \omega = (-1)^{rs} \omega \wedge \omega'$.
\item\ \score{}%12.
Show how to recover the atlas for $G_k(F^n)$ in Lecture~9 (lecture
notes p.19) from the Pl\"ucker coordinates in Lecture~22 (lecture
notes, p.46).
\end{enumerate}
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