This volume presents a course in linear algebra for undergraduate mathematics students. It is considerably wider in its scope than most of the available methods and prepares the students for advanced work in algebra, differential equations, and functional analysis. Therefore, for example, it is transformation-oriented rather than matrix oriented, and whenever possible results are proved for arbitrary vector spaces and not merely for finite-dimensional vector spaces. Also, by proving results for vector spaces over arbitrary fields, rather than only the field of real or complex numbers, it prepares the way for the study of algebraic coding theory, automata theory, and other subjects in theoretical computer science.
Topics are dealt with thoroughly, including ones that normally do not feature in undergraduate textbooks, and many novel and challenging exercises are given. The fact that most students are computer-literate is taken into account, not so much by emphasizing computational aspects of linear algebra which are best left to the computer, but by concentrating on the theory behind it. Audience: Recommended for a one-year undergraduate course in linear algebra.
Preface. Basic notation and terminology. I. Fields and integral domains. II. Vector spaces. III. Linear independence and dimension. IV. Linear transformations. V. Endomorphism rings of vector spaces. VI. Representation of linear transformations by matrices. VII. Rings of square matrices. VIII. Systems of linear equations. IX. Determinants. X. Eigenvectors and eigenvalues. XI. The Jordan canonical form. XII. The dual space. XIII. Inner product spaces. XIV. Endomorphisms of inner product spaces. XV. The Moore-Penrose pseudoinverse. XVI. Bilinear transformations and forms. XVII. Algebras over a field. Index.