  Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite
positive dimension. We consider a pair of linear transformations $A:V\to V$ and
$A^*:V\to V$ that satisfy both conditions below:
  (i) There exists a basis for $V$ with respect to which the matrix
representing $A$ is diagonal and the matrix representing $A^*$ is irreducible
tridiagonal.
  (ii) There exists a basis for $V$ with respect to which the matrix
representing $A^*$ is diagonal and the matrix representing $A$ is irreducible
tridiagonal.
  We call such a pair a {\it Leonard pair} on $V$. Referring to the above
Leonard pair, we investigate 24 bases for $V$ on which the action of $A$ and
$A^*$ takes an attractive form. With respect to each of these bases, the
matrices representing $A$ and $A^*$ are either diagonal, lower bidiagonal,
upper bidiagonal, or tridiagonal.
