Contents 1 Linearity - an informal introduction PART Ⅰ PRELINMINARIES 2 Sets and functions 3 Groups 4 Fields PART Ⅱ VECTOR SPACES 5 Coordinate vectors 6 Vector spaces 7 Elementary vector space properties 8 Vector subspaces PART Ⅲ BASIC GEOMETRY 9 The dot product 10 Vector and triple product 11 Lines and planes PART Ⅳ LINEAR MAPS AND MATRICES 12 Introduction to Linear maps 13 Matrices 14 The structure of linear maps 15 Linear maps in terms of matrices PART Ⅴ LINAR SYSTEMS AND ALGORITHMS 16 Computing with matrices 17 Linear systems 18 Determinants PART Ⅵ EIGENVALUES AND GIGENVECTORS 19 Basic of eigenvalues 20 Diagonalizing linear maps 21 The Jordan normal form PART Ⅶ INNER PRODUCT VECTOR SPACES 22 Scalar products 23 Adjoint and unitary maps 24 Diagonaliztion - again 25 Bi-linear and sesqui-linear forms* PART Ⅷ DUAL AND TENSOR VECTOR SPACES* 26 The dual vector space* 27 Tensors* References Index