Elements of Linear Space

Preface

Part I


1 Real Euclidean Space


1.1 Scalars and Vectors


1.2 Sums and Scalar Multiples of Vectors


1.3 Linear Independence


1.4 Theorem


1.5 Theorem


1.6 Theorem


1.7 Base (Co-Ordinate System)


1.8 Theorem


1.9 Inner Product of Two Vectors


1.10 Projection of a Vector on an Axis


1.11 Theorem


1.12 Theorem


1.13 Theorem


1.14 Orthonormal Base


1.15 Norm of a Vector and Angle Between Two Vectors in Terms of Components


1.16 Orthonormalization of a Base


1.17 Subspaces


1.18 Straight Line


1.19 Plane


1.20 Distance Between a Point and a Plane

Exercises

Additional Problems


2 Linear Transformations and Matrices


2.1 Definition


2.2 Addition and Multiplication of Transformations


2.3 Theorem


2.4 Matrix of a Transformation A


2.5 Unit and Zero Transformation


2.6 Addition of Matrices


2.7 Product of Matrices


2.8 Rectangular Matrices


2.9 Transform of a Vector

Exercises 2

Additional Problems


3 Determinants and Linear Equations


3.1 Definition


3.2 Some properties of Determinants


3.3 Theorem


3.4 Systems of Linear Equations

Exercises 3


4 Special Transformations and their Matrices


4.1 Inverse of a Linear Transformation


4.2 A practical Way of Getting the Inverse


4.3 Theorem


4.4 Adjoint of a Transformation


4.5 Theorem


4.6 Theorem


4.7 Theorem


4.8 Orthogonal (Unitary) Transformations


4.9 Theorem


4.10 Change of Base


4.11 Theorem

Exercises 4

Additional Problems


5 Characteristic Equation of a Transformation and Quadratic Forms


5.1 Characteristic Values and Characteristic Vectors of a Transformation


5.2 Theorem


5.3 Definition


5.4 Theorem


5.5 Theorem


5.6 Special Transformations


5.7 Change of a Matrix to Diagonal Form


5.8 Theorem


5.9 Definition


5.10 Theorem


5.11 Quadratic Forms and their Reduction to Canonical Form


5.12 Reduction to Sum or Differences of Squares


5.13 Simultaneous Reduction of Two Quadratic Forms

Exercises 5

Additional Problems

Part II


6 Unitary Spaces

Introduction


6.1 Scalars, Vectors and Vector Spaces


6.2 Subspaces


6.3 Linear Independence


6.4 Theorem


6.5 Base


6.6 Theorem


6.7 Dimension Theorem


6.8 Inner Product


6.9 Unitary Spaces


6.10 Definition


6.11 Theorem


6.12 Definition


6.13 Theorem


6.14 Definition


6.15 Orthonormalization of a Set of Vectors


6.16 Orthonormal Base


6.17 Theorem

Exercises 6


7 Linear Transformations, Matrices and Determinants


7.1 Definition


7.2 Matrix of a Transformation A


7.3 Addition and Multiplication of Matrices


7.4 Rectangular Matrices


7.5 Determinants


7.6 Rank of a Matrix


7.7 Systems of Linear Equations


7.8 Inverse of a Linear Transformation


7.9 Adjoint of a Transformation


7.10 Unitary Transformation


7.11 Change of Base


7.12 Characteristic Values and Characteristic Vectors of a Transformation


7.13 Definition


7.14 Theorem


7.15 Theorem

Exercises 7


8 Quadratic Forms and Application to Geometry


8.1 Definition


8.2 Reduction of a Quadratic Form to Canonical Form


8.3 Reduction to Sum or Difference of Squares


8.4 Simultaneous Reduction of Two Quadratic Forms


8.5 Homogeneous Coordinates


8.6 Change of Coordinate System


8.7 Invariance of Rank


8.8 Second Degree Curves


8.9 Second Degree Surfaces


8.10 Direction Numbers and Equations of Straight Lines and Planes


8.11 Intersection of a Straight Line and a Quadric


8.12 Theorem


8.13 A Center of a Quadric


8.14 Tangent Plane to a Quadric


8.15 Ruled Surfaces


8.16 Theorem

Exercises 8

Additional Problems


9 Applications and Problem Solving Techniques


9.1 A General Projection


9.2 Intersection of Planes


9.3 Sphere


9.4 A Property of the Sphere


9.5 Radical Axis


9.6 Principal Planes


9.7 Central Quadric


9.8 Quadric of Rank 2


9.9 Quadric of Rank 1


9.10 Axis of Rotation


9.11 Identification of a Quadric


9.12 Rulings


9.13 Locus Problems


9.14 Curves in Space


9.15 Pole and Polar

Exercises 9

Part III


10 Some Algeraic Structures

Introduction


10.1 Definition


10.2 Groups


10.3 Theorem


10.4 Corollary


10.5 Fields


10.6 Examples


10.7 Vector Spaces


10.8 Subspaces


10.9 Examples of Vector Spaces


10.10 Linear Independence


10.11 Base


10.12 Theorem


10.13 Corollary


10.14 Theorem


10.15 Theorem


10.16 Unitary Spaces


10.17 Theorem


10.18 Orthogonality


10.19 Theorem


10.21 Orthogonal Complement of a Subspace

Exercises 10


11 Linear Transformations in General Vector Spaces


11.1 Definitions


11.2 Space of Linear Transformations


11.3 Algebra of Linear Transformations


11.4 Finite-Dimensional Vector Spaces


11.5 Rectangular Matrices


11.6 Rank and Range of a Transformation


11.7 Null Space and Nullity


11.8 Transform of a Vector


11.9 Inverse of a Transformation


11.10 Change of Base


11.11 Characteristic Equation of a Transformation


11.12 Cayley-Hamilton Theorem


11.13 Unitary Spaces and Special Transformations


11.14 Complementary Subspaces


11.15 Projections


11.16 Algebra of Projections


11.17 Matrix of a Projection


11.18 Perpendicular Projection


11.19 Decomposition of Hermitian Transformations

Exercises 11


12 Singular Values and Estemates of Proper Values of Matrices


12.1 Proper Values of a Matrix


12.2 Theorem


12.3 Cartesian Decomposition of a Linear Transformation


12.4 Singular Values of a Transformation


12.5 Theorem


12.6 Theorem


12.7 Theorem


12.8 Theorem


12.9 Theorem


12.10 Lemma


12.11 Theorem


12.12 The Space of n-by-n Matrices


12.13 Hilbert Norm


12.14 Frobenius Norm


12.15 Theorem


12.16 Theorem


12.17 Theorem

Exercises 12

Appendix


1. The Plane


2. Comparison of a Line and a Plane


3. Two Planes


4. Lines and Planes


5. Skew Lines


6. Projection Onto a Plane

Index