Vector Spaces of Finite Dimension

THIS book is based on a course of lectures on linear algebra given to second-year Honours students in the University of Birmingham. A familiarity with the most elementary parts of abstract algebra, together with the properties of matrices and determinants, is assumed. These are summarised briefly in the introductory section, after which the first five chapters give a concise account of the theory of finite-dimensional vector spaces without the use of matrices. In Chapter VI, matrices are introduced, and the methods already developed are applied to them. Most ofthe notations and terminology used are standard. We have borrowed the notation (xl y) from quantum theory for the inner product of two vectors (Chapter V) and in Chapter II we have introduced a new term, implicit rank, for a quantity which plays an important role in the reduction of linear transformations, but does not seem to have previously been named. Over 150 exercises are included in the text.