For over a decade it has been a common observation that a "fog" passes over the course in linear algebra once abstract vector spaces are presented. See [2, 3]. We show how this fog may be cleared by having the students translate "abstract" vector-space problems to isomorphic "concrete" settings, solve the "concrete" problem either by hand or with the use of matrix-crunching software, and then back-translate these concrete solutions to the "abstract" setting for final answer. The method is totally general and not limited to special cases. When the course is taught in this new way, students are able to solve "abstract" vector-space problems clearly, easily, and "foglessly."