Vectors, Pure and Applied: A General Introduction to Linear Algebra

Many books in linear algebra focus purely on getting students through exams, but this text explains both the how and the why of linear algebra and enables students to begin thinking like mathematicians. The author demonstrates how different topics (geometry, abstract algebra, numerical analysis, physics) make use of vectors in different ways and how these ways are connected, preparing students for further work in these areas. The book is packed with hundreds of exercises ranging from the routine to the challenging. Sketch solutions of the easier exercises are available online.

Table of Contents

Part I Familiar vector spaces
1 Gaussian elimination
2 A little geometry
3 The algebra of square matrices
4 The secret life of determinants
5 Abstract vector spaces
6 Linear maps from Fn to itself
7 Distance preserving linear maps
8 Diagonalisation for orthonormal bases
9 Cartesian tensors
10 More on tensors

Part II General vector spaces
11 Spaces of linear maps
12 Polynomials in L(U,U)
13 Vector spaces without distances
14 Vector spaces with distances
15 More distances
16 Quadratic forms and their relatives