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\hfill \textbf{April 14, 2015}

\vspace{0.5 cm} \centerline{ \textbf{\large{Homework 4\hspace{1 cm}
Math 6303}}}

\vspace {1 cm}
\begin{enumerate}

\item Let $m_{\tau}(x)$ be the minimal
polynomial of the linear map $\tau$ on the
finite dimensional vector space $\mbf{V}$.
Show that $\tau$ is bijective iff
$m_{\tau}(0)\ne0$. Conclude that for a
bijective map $\tau$, the map $\tau^{-1}$ is
a polynomial in $\tau$, i.e.,
$\tau^{-1}=p(\tau)$ for some $p(x)\in
\mbf{F}[x]$. \item Let $(\mbf{V},\tau)$ be as
before. Show that $\mbf{V}=\mbf{V}_0\oplus
\mbf{V_1}$ where $\mbf{V}_0$ and $\mbf{V}_1$
are invariant subspaces of $\mbf{V}$ and
where $\tau\upharpoonright \mbf{V}_0$ is
nilpotent, i.e., $\tau^{k}=0$ for some $k$
and where $\tau\upharpoonright \mbf{V}_1$ is
bijective.

\item List all linear maps on $\mbb{R}^{2}$
and on $\mbb{R}^{3}$ according to their
similarity types.
\item Assume that $\tau$ is
diagonalizable. Is it true that every
invariant subspace is diagonalizable? Prove
your answer.
\item A linear operator $T:\mathbb{R}^n\to \mathbb{R}^n$ is symmetric if $(Tv,w)=(v,Tw)$ holds for all $v$ and $w$ in $\mathbb{R}^n$. Show that the minimal polynomial $p_T(x)$ of $T$ is a product
 of $k-$many linear factors $(x-\lambda)$ with real eigenvalues $\lambda$ and that $\mathbb{R}^n$ decomposes into a direct sum of $k-$many eigen-spaces which are pairwise perpendicular. Each eigen-space has of course an orthogonal basis.
 \end{enumerate}
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