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\begin{center}
  {\large Mathematics 111}\\Spring 2007\\Homework 5
\end{center}

\begin{enumerate}
\item Show that a vector $v = (a_1, \dots, a_n) \in \Z^n$ extends to a
  basis $\{v, v_2, \dots, v_n\}$ of $\Z^n$ if and only if the $a_i$
  are coprime, that is $a_1\Z + \cdots + a_n \Z = \Z$.  Hint: For one
  direction, come up with a short exact sequence that splits.

\item Let $A =
  \begin{pmatrix}
    4&7&2\\2&4&6
  \end{pmatrix}$.
  \begin{enumerate}
  \item If $\varphi:\Z^3 \to \Z^2$ is a $\Z$-linear map whose matrix
    with respect to the standard bases is $A$, determine the structure
    of the cokernel $\Z^2/Im(\varphi)$ as a direct sum of cyclic
    groups.  Find a minimal set of generators for this quotient Hint:
    The image of $\varphi$ is the span of the columns (i.e., the
    column space), and you may assume wlog that elementary column
    operations (over $\Z$) leave the column space unchanged.

    Explain how your answer is connected to the elementary divisors
    theorem.

  \item Determine all integer solutions to $ A
    \begin{pmatrix}
      x\\y\\z
    \end{pmatrix} =
    \begin{pmatrix}
      0\\0
    \end{pmatrix}$.  Hint: Elementary row operations (over $\Z$) do
    not change the kernel.


  \end{enumerate}


\end{enumerate}






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