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\begin{document}
\begin{center}
{\bf Homework 10}\\
(due Wednesday, November 16)
\end{center}
\vspace*{.3in}

\noindent{\bf Book Problems:}
\begin{description}
\item[4.1, page 103-104] 1.8, 1.9
\end{description}

\begin{prob}
Let $T : \R^2 \to \R^2$ be the linear transformation whose matrix relative to the
standard basis is 
$A = \left[\begin{matrix} 5 & -5 \\ 1 & 1\end{matrix}\right].$ 
The eigenvalues of $T$ are complex. Find them, and give the diagonalization of the ‘complexified’ transformation 
$T:\C^2 \to \C^2$. Also, imitate the example I did in class by finding a real change of coordinate matrix $C$ and a rotation/scaling matrix $R$ such that $A = CRC^{-1}$. State the rotation angle and scaling factor explicitly.
\end{prob}

\begin{prob}
Let $T:V\to V$ be a linear transformation and $H_1,H_2\subset V$ be
$T$-invariant subspaces.  Prove or give counterexamples to each of the
following assertions.
\begin{enumerate}
\item $H_1\cap H_2$ is a $T$-invariant subspace. 
\item $H_1+H_2$ is a $T$-invariant subspace.
\end{enumerate}
\end{prob}


\begin{prob}
Fix a non-zero number $\lambda\in\R$ and consider the matrix $A = \left[\begin{matrix} \lambda & 1 \\ 0 & \lambda \end{matrix}\right]$ (which is not diagonalizable).  Find (and verify by induction) a formula for $A^n$.  How does this differ from 
$
\left[\begin{matrix} \lambda & 0 \\ 0 & \lambda \end{matrix}\right]^n?
$
In particular, how do the sizes of the largest entries in the matrices compare as $n$ gets large?
\end{prob}

\begin{prob}
Give examples of linear transformations $T:\R^3\to\R^3$ of ranks 0, 1, and 2 such that the characteristic polynomial of $T$ is
$p(\lambda) = \lambda^3$ in all three cases.  Explain why it is impossible to find such an example that has rank 3.
\end{prob}

\begin{prob}
In each of the following, consider the linear operator $T:\R^3\to\R^3$ given by $T(\vx) = A\vx$ where the matrix $A$ is as shown.  Find and describe \emph{all} $T$-invariant subspaces.  In particular, say how many such subspaces there are.
\begin{enumerate}
 \item $A = \left[\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}\right]$;

 \item $A = \left[\begin{matrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{matrix}\right]$;

 \item $A = \left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{matrix}\right]$;
 \item $A = \left[\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{matrix}\right]$
\end{enumerate}
Finally, let
$
C = \left[\begin{matrix} 2 & -1 & 0 \\ 1 & 2 & 1 \\ 0 & 0 & 3 \end{matrix}\right].
$
Choose one of the above matrices $A$ (your pick) and suppose that $T(\vx) = B\vx$ where $B = CAC^{-1}$. 
Then what are the invariant subspaces of $T$? 
\end{prob}



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\end

