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\title{Math 728\\Fall 2004\\Homework 7}
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\centerline{\it Due Monday November 29, 2004.}

\centerline{}

\noindent 1. Find the canonical forms (the rational canonical form, primary rational canonical form and Jordan canonical form if possible) for the following matrices over $\mathbb{Q}$:
\begin{itemize}
\item[a)] (5 pt)
$\left[
\begin {array}{ccccc}
\ \ 3&-4&-6&-9&\ \ 0\\

 -1&\ \ 7&\ \ 8&\ \ 11&-1\\

\ \ 1&\ \ 0&\ \ 1&-1&-1\\

\ \ 0&-3&-4&-4&\ \ 1\\

\ \ 1&-1&-2&-3&\ \ 1
\end {array}
\right]$

\item[b)] (5 pt) 
$\left[
\begin {array}{ccccc}
\ \ 0&\ \ 2&\ \ 0&-2&\ \ 3\\

\ \ 4&-3&-2&-1&-1\\

 -1&\ \ 4&\ \ 0&-2&\ \ 5\\

\ \ 1&\ \ 2&\ \ 0&-3&\ \ 3\\

 -2&\ \ 2&\ \ 1&\ \ 0&\ \ 1
\end {array}
\right]$

\end{itemize}

\centerline{}

\noindent 2. (5 pt) Show that an $n\times n$ matrix ($A$) over a field $\mathbb{F}$ is similar to a diagonal matrix if and only if there is a basis of $\mathbb{F}^n$ consisting of eigenvectors of $A$ (where you need it, you may assume that the eigenvalues of $A$ are all in $\mathbb{F}$...in particular, this holds if $\mathbb{F}$ is algebraically closed).

\centerline{}

\noindent 3. (5 pt) Let $0\neq p\in\mathbb{Z}$ be a prime and $\mathfrak{M}$ be the category of finite abelian $p-$groups. Compute $K_0(\mathfrak{M})$. 

\centerline{}

\noindent 4. (5 pt) Let $\mathfrak{N}$ be a category that is closed under countable direct sum (that is, if $\{C_i\}_{i=1}^\infty$ is a collection of objects of $\mathfrak{N}$, then $\oplus_{i=1}^\infty C_i$ is also an object in $\mathfrak{N}$). Show that $K_0(\mathfrak{N})=0$.



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