Spatial Transformations and Visualization: Selected Common Threads and Root Concepts Linking Old to New

Selected Common Threads and Root Concepts Linking Old to New Sandra L. Arlinghaus* and Joseph Kerski** Introduction In the twentieth century, much of mathematics experienced a well-known paradigm shift in moving from a focus on mathematical objects to a focus on transformations of objects. Thus, one sees, for example, linear algebra courses that focus on matrix vector products as linear transformations, rather than on the algebra of the matrices, themselves. As Birkhoff and Mac Lane note in their ground-breaking mid-20 th Century classic, A Survey of Modern Algebra, “It is desirable to have at hand not merely the formal operations with matrices, but also the (often neglected) interpretation of the matrices by linear transformations” (p. vi). Indeed, midway through this work (pp. 192-194) the authors develop the Gram-Schmidt Orthogonalization Process to show how to orthogonalize any basis of a Euclidean vector space, leading to the result that all finite-dimensional abstract Euclidean vector spaces have thus been determined.