《Elements of Linear Spaces》

PART Ⅰ1

1REAL EUCLIDEAN SPACE1

1．1 Scalars and vectors1

1．2 Sums and scalar multiples of vectors1

1．3 Linear independence2

1．4 Theorem2

1．5 Theorem2

1．6 Theorem2

1．7 Base(Co-ordinate system)3

1．8 Theorem4

1．9 Inner product of two vectors5

1．10 Projection of a vector on an axis5

1．11 Theorem6

1．12 Theorem6

1．13 Theorem7

1．14 Orthonormal base7

1．15 Norm of a vector and angle between two vectors in terms of components7

1．16 Orthonormalization of a base8

1．17 Subspaces9

1．18 Straight line10

1．19 Plane11

1．20 Distance between a point and a plane12

Exercises 113

Additional Problems15

2LINEAR TRANSFORMATIONS AND MATRICES16

2．1 Definition16

2．2 Addition and Multiplication of Transformations16

2．3 Theorem16

2．4 Matrix of a Transformation A16

2．5 Unit and zero transformation19

2．6 Addition of Mat-rices20

2．7 Product of Matrices20

2．8 Rectangular matrices21

2．9 Transform of a vector21

Exercises 223

Additional Problems25

3DETERMINANTS AND LINEAR EQUATIONS28

3．1 Definition28

3．2 Some properties of determinants29

3．3 Theorem29

3．4 Systems of Linear equations29

Exercises 334

4SPECIAL TRANSFORMATIONS AND THEIR MATRICES37

4．1 Inverse of a linear transformation37

4．2 A practical way of getting the inverse38

4．3 Theorem38

4．4 Adjoint of a transformation38

4．5 Theorem38

4．6 Theorem39

4．7 Theorem39

4．8 Orthogonal(Unitary)transformations39

4．9 Theorem40

4．10 Change of Base40

4．11 Theorem41

Exercises 443

Additional Problems44

5CHARACTERISTIC EQUATION OF A TRANSFORMATION AND QUADRATIC FORMS47

5．1 Characteristic values and characteristic vectors of a transformation47

5．2 Theorem47

5．3 Definition48

5．4 Theorem48

5．5 Theorem48

5．6 Special transformations48

5．7 Change of a matrix to diagonal form49

5．8 Theorem50

5．9 Definition51

5．10 Theorem51

5．11 Quadratic forms and their reduction to canonical form52

5．12 Reduction to sum or differences of squares54

5．13 Simultaneous reduction of two quadratic forms54

Exercises 557

Additional Problems58

PART Ⅱ61

6UNITARY SPACES61

Introduction61

6．1 Scalars，Vectors and vector spaces61

6．2 Subspaces61

6．3 Lin-ear independence61

6．4 Theorem61

6．5 Base62

6．6 Theorem62

6．7 Dimension theorem63

6．8 Inner Product63

6．9 Unitary spaces63

6．10 Definition63

6．11 Theorem63

6．12 Definition63

6．13 Theorem63

6．14 Definition64

6．15 Orthonormalization of a set of vectors64

6．16 Orthonormal base64

6．17 Theorem64

Exercises 665

7LINEAR TRANSFORMATIONS，MATRICES AND DETERMINANTS67

7．1 Definition67

7．2 Matrix of a Transfotmation A67

7．3 Addition and Multiplication of Matrices67

7．4 Rectangular matrices68

7．5 Determinants68

7．6 Rank of a matrix69

7．7 Systems of linear equations70

7．8 Inverse of a linear transformation72

7．9 Adjoint of a transformation73

7．10 Unitary Transformation73

7．11 Change of Base74

7．12 Characteristic values and Characteristic vectors of a transformation74

7．13 Definition74

7．14 Theorem75

7．15 Theorem75

Exercises 776

8QUADRATIC FORMS AND APPLICATION TO GEOMETRY79

8．1 Definition79

8．2 Reduction of a quadratic form to canonical form79

8．3 Reduction to Sum or difference of squares80

8．4 Simultaneous reduction of two quadratic forms80

8．5 Homogeneous Coordinates80

8．6 Change of coordinate system80

8．7 Invariance of rank81

8．8 Second degree curves82

8．9 Second degree Surfaces84

8．10 Direction numbers and equations of straight lines and planes89

8．11 Intersection of a straight line and a quadric89

8．12 Theorem90

8．13 A center of a quadric91

8．14 Tangent plane to a quadric92

8．15 Ruled surfaces93

8．16 Theorem95

Exercises 896

Additional Problems97

9APPLICATIONS AND PROBLEM SOLVING TECHNIQUES100

9．1 A general projection100

9．2 Intersection of planes100

9．3 Sphere101

9．4 A property of the sphere101

9．5 Radical axis102

9．6 Principal planes103

9．7 Central quadric104

9．8 Quadric of rank 2105

9．9 Quadric of rank 1106

9．10 Axis of rotation107

9．11 Identification of a quadric107

9．12 Rulings108

9．13 Locus problems108

9．14 Curves in space109

9．15 Pole and polar110

Exercises 9111

PART Ⅲ115

10SOME ALGEBRAIC STRUCTURES115

Introduction115

10．1 Definition115

10．2 Groups115

10．3 Theorem115

10．4 Corollary115

10．5 Fields115

10．6 Examples116

10．7 Vector spaces116

10．8 Subspaces116

10．9 Examples of vector spaces116

10．10 Linear independence117

10．11 Base117

10．12 Theorem117

10．13 Corollary117

10．14 Theorem117

10．15 Theorem118

10．16 Unitary spaces118

10．17 Theorem119

10．18 Orthogonality120

10．19 Theorem120

10．2l Orthogonal complement of a subspace121

Exercises 10121

11LINEAR TRANSFORMATIONS IN GENERAL VECTOR SPACES123

11．1 Definitions123

11．2 Space of linear transformations123

11．3 Algebra of linear transformations123

11．4 Finite-dimensional vector spaces124

11．5 Rectangular matrices124

11．6 Rank and range of a transformation124

11．7 Null space and nullity125

11．8 Trans-form of a vector125

11．9 Inverse of a transformation125

11．10 Change of base126

11．11 Characteristic equation of a transformation126

11．12 Cayley-Hamilton Theorem126

11．13 Unitary spaces and special transformations127

11．14 Complementary subspaces128

11．15 Projections128

11．16 Algebra of projections128

11．17 Matrix of a projection129

11．18 Perpendicular projection129

11．19 Decomposition of Hermitian transformations129

Exercises 11130

12SINGULAR VALUES AND ESTIMATES OF PROPER VALUES OF MATRICES132

12．1 Proper values of a matrix132

12．2 Theorem132

12．3 Cartesian decomposition of a linear transformation133

12．4 Singular values of a transformation134

12．5 Theorem134

12．6 Theorem135

12．7 Theorem135

12．8 Theorem135

12．9 Theorem136

12．10 Lemma136

12．11 Theorem137

12．12 The space of n-by-n matrices137

12．13 Hilbert norm137

12．14 Frobenius norm138

12．15 Theorem138

12．16 Theorem139

12．17 Theorem141

Exercises 12141

APPENDIX143

1．The plane143

2．Comparison of a line and a plane143

3．Two planes144

4．Linesand planes144

5．Skew lines145

6．Projection onto a plane145

Index147