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

\noindent{\bf Book Problems:} 
\begin{description}
\item[4.1, p103] 1.2 (1st and 3rd matrices---you can use matlab or calculators to solve linear systems here), 1.6
\item[4.2, p112-113] 2.4, 2.5 (1st and 3rd matrices only), 2.6ab
\end{description}


\begin{prob} Identify the (real) eigenvalues and corresponding eigenspaces for each of the linear transformations described in Problem 4 from Homework 6. You do not need to justify your answers.
\end{prob}


\begin{prob}
Let $P(\R)$ denote the set of all polynomials with real coefficients and let $T:P(\R)\to P(\R)$ denote the linear transformation $T(p) = q$ where $q(x) = p(x+1)$.  Show that $1$ is the only eigenvalue of $T$ and find its eigenspace.
\end{prob}

\begin{prob} 
In order to buy a house, you borrow 100K from your local bank. The terms are as follows: you pay 6\% interest compounded once a year on the remaining principal, you pay down this principal by making one payment per year of P dollars, and P is chosen so that the loan is repaid at the end of 30 years. What is P?
\end{prob}

\begin{prob}
It’s cold season, and campus splits into two groups of people: those who have
colds and those who don't. On any given day, a healthy person has a 10\% chance of catching
a new cold, and an afflicted person has a 20\% chance of recovering from his or her cold.
Given that cold season lasts a good 100 days, about what are the chances that a given
person will have a cold at the end of the season? Explain why your answer has nothing to
do with whether or not the person starts the season with a cold. (Hint: let $p_n$ be the fraction
of people with colds on day $n$ and $q_n$ the fraction of people without.  What is the probable
relationship between $(p_{n+1}, q_{n+1})$ and $(p_n, q_n)$?)
\end{prob}

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