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\begin{center}
  {\large Mathematics 111}\\Spring 2011\\Homework 3
\end{center}

\begin{enumerate}
\item (Commutative diagrams gone mad) Given a ring $R$ with identity
  and $R$-modules $ A, B, M$, consider the following diagram with
  $R$-linear maps $f, g$:
  \[
  \xymatrix{& B \ar[d]_g \\
    A \ar[r]^f & M }
  \]
  A \textit{pullback} for this diagram (also called a fiber product of
  $f$ and $g$) consists of the following data:

  \begin{compactitem}
  \item A module $X$ and $R$-linear maps $p: X \to A$, $q: X \to B$
    making the following diagram commute.
    \[
    \xymatrix{
      X \ar[d]_p \ar[r]^q & B \ar[d]_g \\
      A \ar[r]^f & M }
    \]
  \item For every $R$-module $X'$ admitting a commutative diagram of
    linear maps (same $f, g$)
    \[
    \xymatrix{
      X' \ar[d]_{p'} \ar[r]^{q'} & B \ar[d]_g \\
      A \ar[r]^f & M }
    \]
    there is a unique $R$-linear map $h : X' \to X$ such that the
    following diagram commutes:
    \[
    \xymatrix{ X' \ar@/_/ @{->} [ddr]_{p'} \ar@{->}[dr]^h
      \ar@/^/[drr]^{q'} \\
      &X \ar[d]^p \ar[r]_q & B \ar[d]_g \\
      & A \ar[r]^f & M }
    \]

    \vfill

    Now to the exercise: Let $X = \{ (a, b) \in A \times B \mid
    f(a) = g(b)\}$, $p$ and $q$ the standard projections to the
    factors $A$ and $B$.  Show that $X$ together with the associated
    data form a pullback, i.e., verify that $X$ is an $R$-module and
    that the above universal mapping property holds for this choice of
    $X$ and maps $p,q$.  \smallskip

    Remark: Dummit and Foote introduce the pullback as a subset of the
    direct sum $A\oplus B$. Categorically, you should find the direct
    product a more useful perspective.
  \end{compactitem}

  \newpage
\item Let $R$ be a ring, and consider two exact sequences of
  $R$-modules
  \[\xymatrix{%
    0 \ar@{>}[r]& K \ar@{>}[r] &P \ar@{>}[r]^\varphi&M\ar@{>}[r] &0& 0
    \ar@{>}[r]& K' \ar@{>}[r] &P' \ar@{>}[r]^{\varphi'}&M\ar@{>}[r] &0
  }
  \]
  where $P$ and $P'$ are projective.  Show that as $R$-modules
  $K' \oplus P \cong K \oplus P'$.  \newline

  \textit{Hint:} Show there is an exact sequence
  \[\xymatrix{%
    0 \ar@{>}[r]& \textrm{ker}\; \pi \ar@{>}[r] &X
    \ar@{>}[r]^\pi&P\ar@{>}[r] &0 }
  \]
  with $\textrm{ker}\; \pi \cong K'$ and where $X$ is the fiber
  product of $\varphi$ and $\varphi'$ as in the first problem.  From
  this deduce that $X \cong K' \oplus P$.  Similarly, show $X \cong K
  \oplus P'$.



\item Let $V$ be a vector space over a field $K$, and let $\varphi: V
  \to V$ be a $K$-linear map.
  \begin{enumerate}
  \item If $V$ is finite dimensional, show that there is a
    positive integer $m$ so that $Im(\varphi^m) \cap
    Ker(\varphi^m) = \{0\}$.

  \item Show by example that this need not hold if $V$ is infinite
    dimensional.
  \end{enumerate}

\item Let $V$ be an $n$-dimensional vector space over a field $K$, and
  $T: V \to V$ a $K$-linear map satisfying $T^n = 0$, but $T^{n-1} \ne
  0$.

  \begin{enumerate}
  \item Show that for $1 \le j \le n$, $rank(T^j):= \dim_k(Im(T^j))$
    equals $n-j$.

  \item Using the preceding part, show that there is a basis of $V$ so
    that the matrix of $T$ with respect to this basis is strictly
    upper triangular, which is to say it has zeros on and below the
    main diagonal.


  \end{enumerate}


\item Let $T : V\to V$ be a linear operator on a complex vector space.
  For two different $T$, we view $V$ as a $\C[x]$-module in the usual
  way.  In each case determine whether or not $V$ is decomposable (as
  a $\C[x]$-module) .  If it is, write $V$ as an appropriate direct
  sum and express the matrix of $T$ with respect to the associated
  basis.  If $V$ is indecomposable, prove it.
  \begin{enumerate}
  \item First consider $T$ whose matrix with respect to a basis
    $B = \{e_1, e_2, e_3\}$ is 
    $\begin{pmatrix}
      3&1&0\\0&3&2\\0&0&3
    \end{pmatrix}.$

\item  Second, consider (a different) $T$ whose matrix with respect to a basis
    $B = \{e_1, e_2, e_3\}$ is 
    $\begin{pmatrix}
      3&0&5\\0&3&0\\0&0&3
    \end{pmatrix}.$
  \end{enumerate}


\end{enumerate}






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