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\begin{document}
\begin{center}
{\bf Homework 11}\\
(due Wednesday, November 30)
\end{center}
\vspace*{.3in}

\noindent{\bf Book Problems:}
\begin{description}
\item[5.1, p122-123] 1.2, 1.3, 1.4 (`$+$' only), 1.5 (in each part of 1.5, give an explicit counterexample to one of the inner product criteria), 1.6.
\item[5.2, p126-127] 2.1 (find an orthogonal basis for this subspace--note that the given vectors are \emph{not} orthogonal), 2.5 (also give the distance of the vector from the subspace), 2.8. 
\item[5.3, p131-132] 3.3, 3.5
\item[5.4, p138] (\emph{These problems delayed til final homework}) 4.1, 4.3
\end{description}





\begin{prob} Prove or give a counterexample to the following assertion: if $T:V\to V$ is a linear operator on a two dimensional vector space, and $H\subset V$ is $T$-invariant, then there is a complementary subspace $H'\subset V$
that is also $T$-invariant.
\end{prob}


\begin{prob}
Let $T : \R^3 \to \R^3$ be given by $T(\vx) = A\vx$ where
$$
A = \left[\begin{matrix} 
           2 & 0 & 1 \\
           1 & -1 & 2 \\
           1 & 3 & 1 
          \end{matrix}\right]
$$
Find the characteristic polynomial for $T$ using the alternative method I described in class on Friday 11/11.
That is, choose a vector $\vv\in\R^3$, and find the matrix for $T$ relative to the basis $\{\vv,T\vv,T^2\vv\}$
for $\R^3$.
\end{prob}


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