FEATURES/BENEFITS *Takes a geometric point of view. *Emphasis on the numerically favored array forms of many algorithms. *Emphasis on equivalence and duality concepts for the solution of several related problems in adaptive filtering, estimation, and control. *These features are generally absent in most prior treatments, ostensibly on the grounds that they are too abstract and complicated. It is the authors hope that these misconceptions will be dispelled by the presentation herein, and that the fundamental simplicity and power of these ideas will be more widely recognized and exploited. Among other things, these features already yielded new insights and new results for linear and nonlinear problems in areas such as adaptive filtering, quadratic control, and estimation, including the recent H? theories.