Courses that study vectors and elementary matrix theory and introduce
linear transformations have proliferated greatly in recent years. Most
of these courses are taught at the undergraduate level as part of, or
adjacent to, the second-year calculus sequence. Although many students
will ultimately find the material in these courses more valuable than
calculus, they often experience a class that consists mostly of learning
to implement a series of computational algorithms. The objective of this
text is to bring a different vision to this course, including many of
the key elements called for in current mathematics-teaching reform
efforts. Three of the main components of this current effort are the
following: 1. Mathematical ideas should be introduced in meaningful
contexts, with after a clear understanding formal definitions and
procedures developed of practical situations has been achieved. 2. Every
topic should be treated from different perspectives, including the
numerical, geometric, and symbolic viewpoints. 3. The important ideas
need to be visited repeatedly throughout the term, with students'
understan9ing deepening each time. This text was written with these
three objectives in mind. The first two chapters deal with situations
requiring linear functions (at times, locally linear functions) or
linear ideas in geometry for their understanding. These situations
provide the context in which the formal mathematics is developed, and
they are returned to with increasing sophistication throughout the text.
