  Sequences of unit vectors for which the Kaczmarz algorithm always converges
in Hilbert space can be characterized in frame theory by tight frames with
constant 1. We generalize this result to the context of frames and bases. In
particular, we show that the only effective sequences which are Riesz bases are
orthonormal bases. Moreover, we consider the infinite system of linear
algebraic equations $A x = b$ and characterize the (bounded) matrices $A$ for
which the Kaczmarz algorithm always converges to a solution.
