  The Kaczmarz algorithm is an iterative method for solving a system of linear
equations. It can be extended so as to reconstruct a vector $x$ in a
(separable) Hilbert space from the inner-products $\{\langle x, \phi_{n}
\rangle\}$. The Kaczmarz algorithms defines a sequence of approximations from
the sequence $\{\langle x, \phi_{n} \rangle\}$; these approximations only
converge to $x$ when $\{\phi_{n}\}$ is ${effective}$. We dualize the Kaczmarz
algorithm so that $x$ can be obtained from $\{\langle x, \phi_{n} \rangle\}$ by
using a second sequence $\{\psi_{n}\}$ in the reconstruction. This allows for
the recovery of $x$ even when the sequence $\{\phi_{n}\}$ is not effective; in
particular, our dualization yields a reconstruction when the sequence
$\{\phi_{n}\}$ is $almost$ $effective$. We also obtain some partial results
characterizing when the sequence of approximations from $\{\langle x, \phi_{n}
\rangle\}$ using $\{\psi_{n}\}$ converges to $x$, in which case $\{(\phi_n,
\psi_n)\}$ is called an $effective$ $pair$.
