Page 11 - Advanced Linear Algebra

Contents

Preface to the Third Edition, vii
Preface to the Second Edition, ix
Preface to the First Edition, xi
Preliminaries, 1
Part 1: Preliminaries, 1
Part 2: Algebraic Structures, 17

Part I—Basic Linear Algebra, 33

1 Vector Spaces, 35
Vector Spaces, 35
Subspaces, 37
Direct Sums, 40
Spanning Sets and Linear Independence, 44
The Dimension of a Vector Space, 48
Ordered Bases and Coordinate Matrices, 51
The Row and Column Spaces of a Matrix, 52
The Complexification of a Real Vector Space, 53
Exercises, 55
2 Linear Transformations, 59
Linear Transformations, 59
The Kernel and Image of a Linear Transformation, 61
Isomorphisms, 62
The Rank Plus Nullity Theorem, 63
Linear Transformations from - to - , 64
Change of Basis Matrices, 65
The Matrix of a Linear Transformation, 66
Change of Bases for Linear Transformations, 68
Equivalence of Matrices, 68
Similarity of Matrices, 70
Similarity of Operators, 71
Invariant Subspaces and Reducing Pairs, 72
Projection Operators, 73