# 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