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 independence 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 theorem 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 Transfotmation 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 Characteristic values and Characteristic vectors of a transformation 74
7．13 Definition 74
7．14 Theorem 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 Homogeneous 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 equations 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 property 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 Corollary 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 Unitary spaces 118
10．17 Theorem 119
10．18 Orthogonality 120
10．19 Theorem 120
10．2l Orthogonal 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．Linesand planes 144
5．Skew lines 145
6．Projection onto a plane 145
Index 147