This text, designed for courses in linear algebra or for supplementary use by students in such courses, possesses the distinct advantage of approaching the subject simultaneously at two levels: the concrete and the axiomatic. Each new property of a vector space is discussed first at one level, then the other. Thus the student is introduced to the elegance and power of mathematical reasoning on the basis of a set of axioms; the gap is bridged between emphasis on problem-solving and the axio...