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\begin{document}
\begin{center}
{\bf Homework 9}\\
(due Wednesday, April 18)
\end{center}
\vspace*{.3in}

\begin{prob}
Suppose $V$ is an inner product space.  Show that any collection $H_1,\dots,H_k\subset V$ of mutually orthogonal subspaces is independent.
\end{prob}

\begin{prob}
Let $V$ be a vector space over $\F$ and $H_1,\dots,H_k\subset V$ be subspaces.  Let $\sigma:H_1\times\dots\times H_k\to V$ be given by $\sigma(\vv_1,\dots,\vv_k) = \vv_1+\dots+\vv_k$.  Show that
\begin{enumerate}
 \item $\sigma$ is injective if and only if $H_1,\dots,H_k$ are independent;
 \item $\sigma$ is surjective if and only if $V = H_1 + \dots +H_k$.
\end{enumerate}
\end{prob}

\begin{prob}
Find $\gcd(3953,1829)$ and express it in the form $3953m + 1829n$, where
$m,n\in\Z$.
\end{prob}

\begin{prob}
Consider the linear transformation 
$T:\R^4\to \R^4$ given by $T(x_1,x_2,x_3,x_4) = (2x_1 + x_2,2x_2,-x_4,x_3)$.
\begin{enumerate}
\item What are all the invariant subspaces of $T$?  Which ones are `minimal'
in the sense that they contain no smaller (non-trivial) $T$-invariant subspaces?
\item What is $p_{T,\vv}$ if $\vv = (0,1,0,0)$?
\item What is $p_{T,\vv}$ if $\vv = (1,0,1,1)$?
\item Is there a vector $\vv\in \R^4$ with $p_{T,\vv}$ equal to the
  characteristic polynomial of $T$?
\end{enumerate}
\end{prob}

\begin{prob}
Give examples of each of the following.
\begin{enumerate}
\item A linear transformation $T:V\to V$ such that $p_{char}\neq p_{T,\vv}$
for any $\vv\in V$.
\item A linear transformation $T:V\to V$ and a vector $\vv\in V$ such that 
$H_{T,\vv}\neq \ker p_{T,\vv}(T)$.
\end{enumerate}
\end{prob}

\begin{prob}
Let us say that a linear transformation $T:V\to V$ is \emph{simple} if the
only $T$-invariant subspaces of $V$ are $V$ and $\{\vzero\}$.  Show that $T$
is simple if and only if its characteristic polynomial $p_{char}$ is 
irreducible.
\end{prob}

\begin{prob}
Suppose that $a,b\in\F[x]$ are relatively prime polynomials and that
$c\in\F[x]$ is a polynomial such that 
$a|bc$.  Prove that $a|c$.  (Hint: use Theorem 3.7 from the classnotes.)
\end{prob}

\begin{prob}
Prove Theorem 3.4 in the classnotes: if $\F$ is a field and $p\in\F[x]$ is 
polynomial of degree at least one, then $p$ has a unique prime factorization.  
(Hint: for existence, use proof by contradiction: suppose that the set $S\subset\F[x]$ of polynomials without prime factorizations is non-empty, and then consider a polynomial $p\in S$ whose degree is as small as possible; for uniqueness, use the previous exercise.)
\end{prob}

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\end

