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La Crosse, thank you for showing me the beauty of mathematics, for encouraging me to pursue graduate school, and for your continued friendship. My final acknowledgements are for my family. Thank you to my parents for exposing me to science and mathematics at a young age, for teaching me to pursue my dreams, and for the endless support. Thank you to my husband, Jordan, who has shown incredible belief and encouragement throughout this process. vi

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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 { x, φn }. The Kaczmarz algorithms defines a sequence of approximations from the sequence { x, φn }; these approximations only converge to x when {φn} is effective. We dualize the Kaczmarz algorithm so that x can be obtained from { x, φn } by using a second sequence {ψn} in the reconstruction. This allows for the recovery of x even when the sequence {φn} is not effective; in particular, our dualization yields a reconstruction when the sequence {φn} is almost effective. We also obtain some partial results characterizing when the sequence of approximations from { x, φn } using {ψn} converges to x, in which case {(φn, ψn)} is called an effective pair.2010 Mathematics Subject Classification. Primary: 41A65, 65D15. Secondary: 42C15, 65F10.

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