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PART Ⅰ 1

1 REAL EUCLIDEAN SPACE 1

1.1 Scalars and vectors 1

1.2 Sums and scalar multiples of vectors 1

1.3 Linear inde-pendence 2

1.4 Theorem 2

1.5 Theorem 2

1.6 Theorem 2

1.7 Base(Co-ordinate system) 3

1.8 Theorem 4

1.9 Inner product of two vectors 5

1.10 Projection of a vector on an axis 5

1.11 Theorem 6

1.12 Theorem 6

1.13 Theorem 7

1.14 Orthonormal base 7

1.15 Norm of a vector and angle between two vectors in terms of components 7

1.16 Orthonormalization of a base 8

1.17 Subspaces 9

1.18 Straight line 10

1.19 Plane 11

1.20 Distance between a point and a plane 12

Exercises 1 13

Additional Problems 15

2 LINEAR TRANSFORMATIONS AND MATRICES 16

2.1 Definition 16

2.2 Addition and Multiplication of Transformations 16

2.3 Theorem 16

2.4 Matrix of a Transformation A 16

2.5 Unit and zero transformation 19

2.6 Addition of Mat-rices 20

2.7 Product of Matrices 20

2.8 Rectangular matrices 21

2.9 Transform of a vector 21

Exercises 2 23

Additional Problems 25

3 DETERMINANTS AND LINEAR EQUATIONS 28

3.1 Definition 28

3.2 Some properties of determinants 29

3.3 Theorem 29

3.4 Systems of linear equations 29

Exercises 3 34

4 SPECIAL TRANSFORMATIONS AND THEIR MATRICES 37

4.1 Inverse of a linear transformation 37

4.2 A practical way of getting the inverse 38

4.3 Theorem 38

4.4 Adjoint of a transformation 38

4.5 Theorem 38

4.6 Theorem 39

4.7 Theorem 39

4.8 Orthogonal(Unitary)transformations 39

4.9 Theorem 40

4.10 Change of Base 40

4.11 Theorem 41

Exercises 4 43

Additional Problems 44

5 CHARACTERISTIC EQUATION OF A TRANSFORMATION AND QUADRATIC FORMS 47

5.1 Characteristic values and characteristic vectors of a transformation 47

5.2 Theorem 47

5.3 Definition 48

5.4 Theorem 48

5.5 Theorem 48

5.6 Special transformations 48

5.7 Change of a matrix to diagonal form 49

5.8 Theorem 50

5.9 Definition 51

5.10 Theorem 51

5.11 Quadratic forms and their reduction to canonical form 52

5.12 Reduction to sum or differences of squares 54

5.13 Simultaneous reduction of two quadratic forms 54

Exercises 5 57

Additional Problems 58

PART Ⅱ 61

6 UNITARY SPACES 61

Introduction 61

6.1 Scalars,Vectors and vector spaces 61

6.2 Subspaces 61

6.3 Lin-ear independence 61

6.4 Theorem 61

6.5 Base 62

6.6 Theorem 62

6.7 Dimension the-orem 63

6.8 Inner Product 63

6.9 Unitary spaces 63

6.10 Definition 63

6.11 Theorem 63

6.12 Definition 63

6.13 Theorem 63

6.14 Definition 64

6.15 Orthonormalization of a set of vectors 64

6.16 Orthonormal base 64

6.17 Theorem 64

Exercises 6 65

7 LINEAR TRANSFORMATIONS,MATRICES AND DETERMINANTS 67

7.1 Definition 67

7.2 Matrix of a Transformation A 67

7.3 Addition and Multiplication of Matrices 67

7.4 Rectangular matrices 68

7.5 Determinants 68

7.6 Rank of a matrix 69

7.7 Systems of linear equations 70

7.8 Inverse of a linear transformation 72

7.9 Adjoint of a transformation 73

7.10 Unitary Transformation 73

7.11 Change of Base 74

7.12 Character-istic values and Characteristic vectors of a transformation 74

7.13 Definition 74

7.14 The-orem 75

7.15 Theorem 75

Exercises 7 76

8 QUADRATIC FORMS AND APPLICATION TO GEOMETRY 79

8.1 Definition 79

8.2 Reduction of a quadratic form to canonical form 79

8.3 Reduction to Sum or difference of squares 80

8.4 Simultaneous reduction of two quadratic forms 80

8.5 Ho-mogeneous Coordinates 80

8.6 Change of coordinate system 80

8.7 Invariance of rank 81

8.8 Second degree curves 82

8.9 Second degree Surfaces 84

8.10 Direction numbers and equa-tions of straight lines and planes 89

8.11 Intersection of a straight line and a quadric 89

8.12 Theorem 90

8.13 A center of a quadric 91

8.14 Tangent plane to a quadric 92

8.15 Ruled surfaces 93

8.16 Theorem 95

Exercises 8 96

Additional Problems 97

9 APPLICATIONS AND PROBLEM SOLVING TECHNIQUES 100

9.1 A general projection 100

9.2 Intersection of planes 100

9.3 Sphere 101

9.4 A prop-erty of the sphere 101

9.5 Radical axis 102

9.6 Principal planes 103

9.7 Central quadric 104

9.8 Quadric of rank 2 105

9.9 Quadric of rank 1 106

9.10 Axis of rotation 107

9.11 Identification of a quadric 107

9.12 Rulings 108

9.13 Locus problems 108

9.14 Curves in space 109

9.15 Pole and polar 110

Exercises 9 111

PART Ⅲ 115

10 SOME ALGEBRAIC STRUCTURES 115

Introduction 115

10.1 Definition 115

10.2 Groups 115

10.3 Theorem 115

10.4 Cor-ollary 115

10.5 Fields 115

10.6 Examples 116

10.7 Vector spaces 116

10.8 Subspaces 116

10.9 Examples of vector spaces 116

10.10 Linear independence 117

10.11 Base 117

10.12 Theorem 117

10.13 Corollary 117

10.14 Theorem 117

10.15 Theorem 118

10.16 Uni-tary spaces 118

10.17 Theorem 119

10.18 Orthogonality 120

10.19 Theorem 120

10.21 Or-thogonal complement of a subspace 121

Exercises 10 121

11 LINEAR TRANSFORMATIONS IN GENERAL VECTOR SPACES 123

11.1 Definitions 123

11.2 Space of linear transformations 123

11.3 Algebra of linear transformations 123

11.4 Finite-dimensional vector spaces 124

11.5 Rectangular matrices 124

11.6 Rank and range of a transformation 124

11.7 Null space and nullity 125

11.8 Trans-form of a vector 125

11.9 Inverse of a transformation 125

11.10 Change of base 126

11.11 Characteristic equation of a transformation 126

11.12 Cayley-Hamilton Theorem 126

11.13 Unitary spaces and special transformations 127

11.14 Complementary subspaces 128

11.15 Projections 128

11.16 Algebra of projections 128

11.17 Matrix of a projection 129

11.18 Perpendicular projection 129

11.19 Decomposition of Hermitian transformations 129

Exercises 11 130

12 SINGULAR VALUES AND ESTIMATES OF PROPER VALUES OF MATRICES 132

12.1 Proper values of a matrix 132

12.2 Theorem 132

12.3 Cartesian decomposition of a linear transformation 133

12.4 Singular values of a transformation 134

12.5 Theorem 134

12.6 Theorem 135

12.7 Theorem 135

12.8 Theorem 135

12.9 Theorem 136

12.10 Lemma 136

12.11 Theorem 137

12.12 The space of n-by-n matrices 137

12.13 Hilbert norm 137

12.14 Frobenius norm 138

12.15 Theorem 138

12.16 Theorem 139

12.17 Theorem 141

Exercises 12 141

APPENDIX 143

1.The plane 143

2.Comparison of a line and a plane 143

3.Two planes 144

4.Lines and planes 144

5.Skew lines 145

6.Projection onto a plane 145

Index 147