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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 algorithm defines a sequence of approximations from the sequence $\{ \langle x, \phi_{n} \rangle \}$; these approximations only converge to $x$ when $\{ \phi_{n} \}$ is \emph{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 \emph{almost effective}. We also obtain some partial results characterizing when the sequence of approximations from $\{\langle \vec{x}, \phi_{n} \rangle\}$ converges to $x$, in which case $\{ (\phi_{n}, \psi_{n}) \}$ is called an effective pair.