Vectors, Matrices and Geometry

Cover

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Title Page, Copyright

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CONTENT

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pp. v-vi

PREFACE

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pp. vii-viii

With the present volume the 3-book series on elementary mathematics is now complete. Like its two predecessors, Fundamental Concepts of Mathematics and Polynomials and Equations, the present book is addressed to a large readership comprising Sixth Form students, first-year undergraduates and students of Institutes of Education. In Chapter One vectors in the plane are simply defined as ordered ...

CHAPTER ONE: VECTORS AND GEOMETRY IN THE PLANE

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pp. 2-38

In school geometry, points on the plane are represented by pairs of real numbers which are called coordinates, and algebraic operations are carried out on the individual coordinates to discover properties of geometric configurations. For example, given two points P and Q represented by (a, b) and (c, d) respectively, the straight line ...

CHAPTER TWO: VECTORS AND GEOMETRY IN SPACE

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pp. 39-89

In this chapter we follow the pattern of last chapter to study algebra of vectors in space and solid geometry side by side. Readers will find a fairly complete treatment of the vector space R³ where most of the important topics are discussed. In spite of the extensive subject of geometry in space, we are only able to include some general algebraic methods in the treatment of lines and planes and a very ...

CHAPTER THREE: CONIC SECTIONS

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pp. 90-166

The curves known as conic sections comprise the ellipse, hyperbola and parabola. They are, after the circle, the simplest curves. This being so, it is not surprising that they have been known and studied for a long time. Their discovery is attributed to Menaechmus, a Greek geometer and astronomer of the 4th century BC. Like Hipprocrates of the 5th century BC before him, Menaechmus, in ...

CHAPTER FOUR: QUADRIC SURFACES

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pp. 167-206

On the plane a linear equation in two variables defines a line. In space a linear equation in three variables defines a plane. A line in space is the intersection of two planes; it is therefore defined by two linear equations in three variables. A quadratic equation in two variables defines a quadratic curve on the plane. Quadratic curves on the plane are called conics because they are plane sections of a ...

CHAPTER FIVE: HIGHER DIMENSIONAL VECTOR SPACES

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pp. 207-242

In the first two chapters, we have learnt the language and techniques of linear algebra of vectors in R² and R³. Instead of moving up one dimension from R³ to R⁴, we shall study the general n-dimensional vector space Rⁿ for any positive integer n. However the nature of the present course only allows us a restricted scope of study. We shall therefore concentrate on the notions of linear independence ...

CHAPTER SIX: MATRIX AND DETERMINANT

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pp. 243-280

Matrices are introduced in the last chapter as a systematic way of presenting the components of m vectors of Rⁿ so that we can keep track of certain calculations being carried out on them. The chief concern of such calculations is to evaluate the rank of a matrix and to select linearly independent row vectors. In this chapter matrices are treated as individual algebraic entities ...

CHAPTER SEVEN: LINEAR EQUATIONS

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pp. 290281-314

In this final chapter, we shall apply results of the last two chapters to investigate systems of linear equations in several unknowns. A necessary and sufficient condition will be given in terms of the ranks of certain matrices and a general method of solution is described. Readers will find that this method is essentially the classical successive eliminations of unknowns but given in terms of elementary row ...

NUMERICAL ANSWERS TO EXERCISES

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pp. 315-340

INDEX

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pp. 341-348