This \book" grew out of a series of twenty five lecture notes for a sophomore linear algebra class taught at the University of California, Davis. The audience was primarily engineering students and students of pure sciences, some of whom may go on to major in mathematics. It was motivated by the lack of a book that taught students basic structures of linear algebra without overdoing mathematical rigor or becoming a mindless exercise in crunching recipes at the cost of fundamental understanding. In particular we wanted a book that was suitable for all students, not just math majors, that focussed on concepts and developing the ability to think in terms of abstract structures in order to address the dizzying array of seemingly disparate applications that can all actually be addressed with linear algebra methods.