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This is a tutorial article on the application of geometrical vector space concepts for deriving the rapidly converging, reduced computation structures known as fast recursive least squares (RLS) adaptive filters. Since potential applications of fast RLS, such as speech coding  and echo, cancellation , have been previously examined in the ASSP Magazine, this article focuses instead on an intuitive geometrical approach to deriving these fast RLS filters for linear prediction. One purpose of this article is to keep the required mathematics at a minimum and instead highlight the properties of the fast RLS filters through geometrical interpretation. The geometrical vector space concepts in this article are then applied to deriving the very important fast RLS structure known as the fast transversal filter (FTF).