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\begin{center}
{\large\textbf{Dartmouth College}\\}
Mathematics 101\\\medskip Homework 1 \small{(due Wednesday, October
  3)}
\end{center}

Let $\Fq$ denote the finite field with $q$ elements.
\begin{enumerate}

\item Show that the group $GL_n(\Fq)$ ($n\ge1$) has cardinality
  $(q^n-1)(q^n-q)\cdots (q^n - q^{n-1})$.  \textit{Hint:} You may use
  without proof that an $n\times n$ matrix is invertible if and only
  if its rows (resp. columns) are linearly independent.

\item Find a formula for the number of $k$-dimensional subspaces of
  the $n$-dimensional vector space $\Fq^n$.  Prove that the number of
  $k$-dimensional subspaces is the same as the number of
  $(n-k)$-dimensional subspaces. In class, we will consider this
  duality for finite dimensional vector spaces over an arbitrary
  field.

\item Let $V$ be an $n$-dimensional vector space over a field $K$, and
  $T: V \to V$ a linear operator such that $T^n = 0$, but $T^{n-1} \ne
  0$.
  \begin{enumerate}
  \item For each integer $j$ with $1 \le j \le n$ show that the rank
    of $T^j$ is $n-j$.

  \item Use this fact to show that there is a basis of $V$ with
    respect to which the matrix of $T$ is strictly upper triangular
    (i.e. all entries on and below the diagonal are zero).

  \end{enumerate}


\item Let $V$ be an $n$-dimensional vector space over a field $K$, and
  $V^* = Hom_k(V, k)$ its dual.  For a subspace $W \subset V$, let
  $W^0$ be the annihilator of W, that is
  \[
  W^0 = \{ \varphi \in V^* \mid \varphi(w) = 0 \textrm{ for all } w
  \in W\}.
  \]
  For subspaces $W_1$ and $W_2$ of $V$, show that
  \begin{enumerate}
  \item $\ds (W_1 + W_2)^0 = W_1^0 \cap W_2^0$.

  \item $\ds (W_1 \cap W_2)^0 = W_1^0 + W_2^0$.

  \end{enumerate}

\end{enumerate}


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