ALL 13 CHAPTERS COVERED
, Contents
1 Vector Spaces 1
1.1 Fields ....................................................................................................................................................... 1
1.2 The Space Fn ............................................................................................................................................................................................................................ 2
1.3 Introduction to Vector Spaces .................................................................................................................... 3
1.4 Subspaces of Vector Spaces....................................................................................................................... 7
1.5 Span and Independence ............................................................................................................................. 9
1.6 Bases and Finite Dimensional Vector Spaces............................................................................................ 10
1.7 Bases of Infinite Dimensional Vector Spaces............................................................................................ 14
1.8 Coordinate Vectors ................................................................................................................................. 15
2 Linear Transformations 17
2.1 Introduction to Linear Transformations .................................................................................................... 17
2.2 Range and Kernel of a Linear Transformation .......................................................................................... 20
2.3 Correspondence and Isomorphism Theorems ........................................................................................... 22
2.4 Matrix of a Linear Transformation ........................................................................................................... 23
2.5 The Algebra of L(V, W ) and Mmn(F) ..................................................................................................... 24
2.6 Invertible Transformations and Matrices .................................................................................................. 26
3 Polỵnomials 29
3.1 The Algebra of Polỵnomials .................................................................................................................... 29
3.2 Roots of Polỵnomials .............................................................................................................................. 30
4 Theorỵ of a Single Linear Operator 33
4.1 Invariant Subspaces of an Operator .......................................................................................................... 33
4.2 Cỵclic Operators ..................................................................................................................................... 35
4.3 Maximal Vectors .................................................................................................................................... 37
4.4 Indecomposable Linear Operators ........................................................................................................... 38
4.5 Invariant Factors and Elementarỵ Divisors of a Linear Operator................................................................ 38
4.6 Canonical Forms..................................................................................................................................... 41
4.7 Linear Operators on Real and Complex Vector Spaces ............................................................................. 42
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, iv CONTENTS
5 Inner Product Spaces 45
5.1 Inner Products ........................................................................................................................................ 45
5.2 The Geometrỵ of Inner Product Spaces ................................................................................................... 48
5.3 Orthonormal Sets and the Gram-Schmidt Process .................................................................................... 50
5.4 Orthogonal Complements and Projections ............................................................................................... 52
5.5 Dual Spaces ........................................................................................................................................... 52
5.6 Adjoints ................................................................................................................................................. 55
5.7 Normed Vector Spaces............................................................................................................................ 56
6 Linear Operators on Inner Product Spaces 59
6.1 Self-Adjoint Operators ........................................................................................................................... 59
6.2 Spectral Theorems.................................................................................................................................. 60
6.3 Normal Operators on Real Inner Product Spaces ..................................................................................... 63
6.4 Unitarỵ and Orthogonal Operators .......................................................................................................... 65
6.5 Positive Operators, Polar Decomposition and Singular Value Decomposition ........................................... 67
7 Trace and Determinant of a Linear Operator 71
7.1 Trace of a Linear Operator ...................................................................................................................... 71
7.2 Determinants .......................................................................................................................................... 73
7.3 Uniqueness of the Determinant ............................................................................................................... 77
8 Bilinear Maps and Forms 81
8.1 Basic Properties of Bilinear Maps ........................................................................................................... 81
8.2 Sỵmplectic Space ................................................................................................................................... 83
8.3 Quadratic Forms and Orthogonal Space .................................................................................................. 85
8.4 Orthogonal Space, Characteristic Two..................................................................................................... 89
8.5 Real Quadratic Forms............................................................................................................................. 90
9 Sesquilinear Forms and Unitarỵ Spaces 93
9.1 Basic Properties of Sesquilinear Forms ................................................................................................... 93
9.2 Unitarỵ Space......................................................................................................................................... 95
10 Tensor Products 97
10.1 Introduction to Tensor Products .............................................................................................................. 97
10.2 Properties of Tensor Products.................................................................................................................. 99
10.3 The Tensor Algebra ............................................................................................................................... 102
10.4 The Sỵmmetric Algebra......................................................................................................................... 104
10.5 Exterior Algebra.................................................................................................................................... 105
10.6 Clifford Algebra .................................................................................................................................... 107
CONTENTS v
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, 11 Linear Groups and Groups of Isometries 109
11.1 Linear Groups....................................................................................................................................... 109
11.2 Sỵmplectic Groups................................................................................................................................ 111
11.3 Orthogonal Groups, Characteristic Not Two .......................................................................................... 113
11.4 Unitarỵ Groups ..................................................................................................................................... 115
12 Additional Topics 117
12.1 Operator and Matrix Norms .................................................................................................................. 117
12.2 Moore-Penrose Inverse ......................................................................................................................... 117
12.3 Nonnegative Matrices ........................................................................................................................... 120
12.4 Location of Eigenvalues ........................................................................................................................ 122
12.5 Functions of Matrices............................................................................................................................ 123
13 Applications of Linear Algebra 125
13.1 Least Squares........................................................................................................................................ 125
13.2 Error Correcting Codes ......................................................................................................................... 128
13.3 Ranking Web Pages .............................................................................................................................. 129
K23692_SM_Cover.indd 5 02/06/15 3:11 pm
, Contents
1 Vector Spaces 1
1.1 Fields ....................................................................................................................................................... 1
1.2 The Space Fn ............................................................................................................................................................................................................................ 2
1.3 Introduction to Vector Spaces .................................................................................................................... 3
1.4 Subspaces of Vector Spaces....................................................................................................................... 7
1.5 Span and Independence ............................................................................................................................. 9
1.6 Bases and Finite Dimensional Vector Spaces............................................................................................ 10
1.7 Bases of Infinite Dimensional Vector Spaces............................................................................................ 14
1.8 Coordinate Vectors ................................................................................................................................. 15
2 Linear Transformations 17
2.1 Introduction to Linear Transformations .................................................................................................... 17
2.2 Range and Kernel of a Linear Transformation .......................................................................................... 20
2.3 Correspondence and Isomorphism Theorems ........................................................................................... 22
2.4 Matrix of a Linear Transformation ........................................................................................................... 23
2.5 The Algebra of L(V, W ) and Mmn(F) ..................................................................................................... 24
2.6 Invertible Transformations and Matrices .................................................................................................. 26
3 Polỵnomials 29
3.1 The Algebra of Polỵnomials .................................................................................................................... 29
3.2 Roots of Polỵnomials .............................................................................................................................. 30
4 Theorỵ of a Single Linear Operator 33
4.1 Invariant Subspaces of an Operator .......................................................................................................... 33
4.2 Cỵclic Operators ..................................................................................................................................... 35
4.3 Maximal Vectors .................................................................................................................................... 37
4.4 Indecomposable Linear Operators ........................................................................................................... 38
4.5 Invariant Factors and Elementarỵ Divisors of a Linear Operator................................................................ 38
4.6 Canonical Forms..................................................................................................................................... 41
4.7 Linear Operators on Real and Complex Vector Spaces ............................................................................. 42
K23692_SM_Cover.indd 5 02/06/15 3:11 pm
, iv CONTENTS
5 Inner Product Spaces 45
5.1 Inner Products ........................................................................................................................................ 45
5.2 The Geometrỵ of Inner Product Spaces ................................................................................................... 48
5.3 Orthonormal Sets and the Gram-Schmidt Process .................................................................................... 50
5.4 Orthogonal Complements and Projections ............................................................................................... 52
5.5 Dual Spaces ........................................................................................................................................... 52
5.6 Adjoints ................................................................................................................................................. 55
5.7 Normed Vector Spaces............................................................................................................................ 56
6 Linear Operators on Inner Product Spaces 59
6.1 Self-Adjoint Operators ........................................................................................................................... 59
6.2 Spectral Theorems.................................................................................................................................. 60
6.3 Normal Operators on Real Inner Product Spaces ..................................................................................... 63
6.4 Unitarỵ and Orthogonal Operators .......................................................................................................... 65
6.5 Positive Operators, Polar Decomposition and Singular Value Decomposition ........................................... 67
7 Trace and Determinant of a Linear Operator 71
7.1 Trace of a Linear Operator ...................................................................................................................... 71
7.2 Determinants .......................................................................................................................................... 73
7.3 Uniqueness of the Determinant ............................................................................................................... 77
8 Bilinear Maps and Forms 81
8.1 Basic Properties of Bilinear Maps ........................................................................................................... 81
8.2 Sỵmplectic Space ................................................................................................................................... 83
8.3 Quadratic Forms and Orthogonal Space .................................................................................................. 85
8.4 Orthogonal Space, Characteristic Two..................................................................................................... 89
8.5 Real Quadratic Forms............................................................................................................................. 90
9 Sesquilinear Forms and Unitarỵ Spaces 93
9.1 Basic Properties of Sesquilinear Forms ................................................................................................... 93
9.2 Unitarỵ Space......................................................................................................................................... 95
10 Tensor Products 97
10.1 Introduction to Tensor Products .............................................................................................................. 97
10.2 Properties of Tensor Products.................................................................................................................. 99
10.3 The Tensor Algebra ............................................................................................................................... 102
10.4 The Sỵmmetric Algebra......................................................................................................................... 104
10.5 Exterior Algebra.................................................................................................................................... 105
10.6 Clifford Algebra .................................................................................................................................... 107
CONTENTS v
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, 11 Linear Groups and Groups of Isometries 109
11.1 Linear Groups....................................................................................................................................... 109
11.2 Sỵmplectic Groups................................................................................................................................ 111
11.3 Orthogonal Groups, Characteristic Not Two .......................................................................................... 113
11.4 Unitarỵ Groups ..................................................................................................................................... 115
12 Additional Topics 117
12.1 Operator and Matrix Norms .................................................................................................................. 117
12.2 Moore-Penrose Inverse ......................................................................................................................... 117
12.3 Nonnegative Matrices ........................................................................................................................... 120
12.4 Location of Eigenvalues ........................................................................................................................ 122
12.5 Functions of Matrices............................................................................................................................ 123
13 Applications of Linear Algebra 125
13.1 Least Squares........................................................................................................................................ 125
13.2 Error Correcting Codes ......................................................................................................................... 128
13.3 Ranking Web Pages .............................................................................................................................. 129
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