Elements of Linear Space
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Elements of Linear Space is a detailed treatment of the elements of linear spaces, including real spaces with no more than three dimensions and complex n-dimensional spaces. The geometry of conic sections and quadric surfaces is considered, along with algebraic structures, especially vector spaces and transformations. Problems drawn from various branches of geometry are given. Comprised of 12 chapters, this volume begins with an introduction to real Euclidean space, followed by a discussion on linear transformations and matrices. The addition and multiplication of transformations and matrices are given emphasis. Subsequent chapters focus on some properties of determinants and systems of linear equations; special transformations and their matrices; unitary spaces; and some algebraic structures. Quadratic forms and their applications to geometry are also examined, together with linear transformations in general vector spaces. The book concludes with an evaluation of singular values and estimates of proper values of matrices, paying particular attention to linear transformations always on a unitary space of dimension n over the complex field. This book will be of interest to both undergraduate and more advanced students of mathematics.
Table of Contents
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
Product details
- No. of pages: 160
- Language: English
- Copyright: © Pergamon 1962
- Published: January 1, 1962
- Imprint: Pergamon
- eBook ISBN: 9781483279091