Full Description
In algebra, an entity is called linear if it can be expressed in terms of addition, and multiplication by a scalar; a linear expression is a sum of scalar multiples of the entities under consideration. Also, an operation is called linear if it preserves addition, and multiplication by a scalar. For example, if A and Bare 2 x 2 real matrices, v is a (row) vector in the real plane, and c is a real number, then v(A + B) = vA + vB and (cv)A = c(vA), that is, the process of applying a matrix to a vector is linear. Linear Algebra is the study of properties and systems which preserve these two operations, and the following pages present the basic theory and results of this important branch of pure mathematics. There are many books on linear algebra in the bookshops and libraries of the world, so why write another? A number of excellent texts were written about fifty years ago (see the bibliography); in the intervening period the 'style' of math ematical presentation has changed. Also, some of the more modern texts have concentrated on applications both inside and outside mathematics. There is noth ing wrong with this approach; these books serve a very useful purpose. But linear algebra contains some fine pure mathematics and so a modern text taking the pure mathematician's viewpoint was thought to be worthwhile.
Contents
1 — Algebraic Preamble.- Groups, Rings and Fields.- Permutation Groups.- Problems 1.- 2 — Vector Spaces and Linear Maps.- Vector Spaces and Algebras.- Bases and Dimension.- Linear Maps.- Direct Sums.- Addendum - Modules.- Problems 2.- 3 — Matrices, Determinants and Linear Equations.- Matrices.- Determinants.- Systems of Linear Equations.- Problems 3.- 4 — Cayley-Hamilton Theorem and Jordan Form.- Polynomials.- Cayley-Hamilton and Spectral Theorems.- Jordan Form.- Problems 4.- 5 — Interlude on Finite Fields.- Finite Fields.- Applications - Linear Codes and Finite Matrix Groups.- Problems 5.- 6 — Hermitian and Inner Product Spaces.- Hermitian and Inner Products, and Norms.- Unitary and Self-adjoint Maps.- Orthogonal and Symmetric Maps.- Problems 6.- 7 — Selected Topics.- The Geometry of Real Quadratic Forms.- Normed Algebras, Quaternions and Cayley Numbers.- to the Representation of Finite Groups.- Problems 7.- Appendix A — Set Theory.- Sets and Maps.- Problems A.- Appendix B — Answers and Solutions to the Problems.- Notation Index.- Definition Index.- Theorem Index.
