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                                %transformation #1 wrt two bases #2,
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\begin{document}
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\begin{center}
  {\large Mathematics 101}\\Fall 2014\\Homework 1
\end{center}


\begin{enumerate}
\item Let $V$ be a finite dimensional vector space over a field $k$,
  and let $\varphi: V\to V$ be $k$-linear.
  \begin{enumerate}

  \item Show that there is a positive integer $m$ so that
    $Im(\varphi^m) \cap ker(\varphi^m) = \{0\}$.

  \item Now suppose that $\varphi^2 = 0$.  Show that the rank of
    $\varphi$ is at most $\dim V/2$.
  \end{enumerate}

\item Let $V$ be a finite dimensional vector space over a field $k$,
  and let $\varphi: V\to V$ be $k$-linear, and suppose that $\varphi^2
  = \varphi$, that is, $\varphi$ is an idempotent map.
  \begin{enumerate}

  \item Show that $V = ker(\varphi) \oplus Im(\varphi)$.
  \item Show that there is a basis of $V$ so that the matrix with
    respect to this basis is diagonal all of whose entries are 0 or 1.
   
  \end{enumerate}


\item Let $V$ be an arbitrary vector space over field $k$, and
  $\varphi$ a linear operator on $V$.  Let $W$ be a subspace which is
  invariant under $\varphi$.  Consequently there are induced maps,
  $\varphi |_W: W \to W$ and $\overline\varphi: V/W \to V/W$, the
  later defined by $\overline\varphi(v+W) = \varphi(v) + W$.


  \begin{enumerate}
  \item Show that if $\varphi |_W$ and $\overline \varphi$ are
    nonsingular (i.e., injective), then $\varphi$ is nonsingular.

  \item Show that the converse holds if $V$ has finite dimension, and
    find a counterexample with $V$ infinite dimensional.
  \end{enumerate}

\item Let $V$ be a vector space over a field $k$, and $\varphi$ a
  linear operator on $V$.  Suppose that $\lambda_1, \dots, \lambda_r$
  are distinct eigenvalues of $\varphi$.  For an eigenvalue $\lambda$,
  denote by $E_\lambda$ the corresponding eigenspace, i.e., $E_\lambda
  = \{ v\in V \mid \varphi(v) = \lambda v\}$.

  \begin{enumerate}
  \item Show that $\ds\sum_{i=1}^r E_{\lambda_i} = \bigoplus_{i=1}^r
    E_{\lambda_i}.$ Hint: It suffices to show that $\ds E_{\lambda_1}
    \cap \sum_{i=2}^r E_{\lambda_i} = \{0\}$.
  \item Conclude that any linear transformation on a finite
    dimensional vector space has at most $\dim (V)$ distinct
    eigenvalues.
  \end{enumerate}

\item Let $\alpha, \beta, \gamma$ be nonzero real numbers, and let $T:
  \R^3 \to \R^3$ be the orthogonal projection onto the subspace $W =
  \{(x,y,z) \mid \alpha x + \beta y + \gamma z = 0\}$. Find the matrix
  of $T$ with respect to the standard ordered basis of $\R^3$.  Hint:
  It would be productive to find the matrix of $T$ with respect to a
  natural basis for the problem, and then use change of basis matrices
  to achieve the desired result.

  \vfill \rightline{(continued on next page)} \newpage

\item Let $k$ be a field and $P \in GL_m(k)$.
  \begin{enumerate}
  
  \item Given a basis $\cC$ for $k^m$, show there is a unique basis
    $\cB$ so that $P = \phantom{}_\cC[Id]_\cB$.  % = \M{Id}{\cB}{\cC}
  \item Given a basis $\cB$ for $k^m$, show there is a unique basis
    $\cC$ so that $P = \phantom{}_\cC[Id]_\cB$.
  \item Let $A$ be an $m\times m$ matrix with entries from $k$.  Show
    there are matrices $P,Q \in GL_m(k)$ with $PAQ$ a diagonal matrix
    with zeros or ones on the diagonal.


  \end{enumerate}
\item Something similar is true over $\Z$ as well.  For $A \in
  M_m(\Z)$, there exist $P,Q \in GL_m(\Z)$ so that $PAQ = \diag(d_1,
  d_2, \dots, d_m)$ with $d_1 \mid d_2\mid \cdots \mid d_m$.  This is
  called the Smith Normal form of a matrix.  Remember that
  multiplication by $P$ and $Q$ correspond to elementary row and
  column operations of the matrix.  Use this observation to analyze
  the following situation.

  Let $M$ be the $\Z$-module $\Z^2$ and $N$ the submodule generated by
  $\col{2}{4}, \col{8}{10}$.  The quotient module $M/N$ is a finitely
  generated abelian group; indeed in this case a finite abelian group.
  Write it as a product of cyclic groups.  Hint: Define $\varphi: \Z^2
  \to \Z^2$ by $\varphi\col01 = \col24$, and $\varphi\col10 =
  \col{8}{10}.$ Then $Im(\varphi) = N$.  Let $\cB=\{e_1=\col01,
  e_2=\col10\}$, $\cC=\{f_1=\col01, f_2=\col10\}$ be the standard
  ordered basis of $\Z^2$ (one for the domain, the other for the
  codomain).  Then $\M{\varphi}{\cB}{\cC} = \left(
    \begin{smallmatrix}
      2&8\\4&10
    \end{smallmatrix}\right)
  $.  Now consider how changing the bases in domain and codomain
  affect $Im(\varphi)$.

\end{enumerate}


\end{document}
