Linear Algebra and Geometry

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PREFACE

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pp. v-vi

Linear algebra is now included in the undergraduate curriculum of most universities. It is generally recognized that this branch of algebra, being less abstract and directly motivated by geometry, is easier to understand than some other branches and that because of the wide applications it should be taught as early as possible. The ...

CONTENTS

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pp. vii-ix

CHAPTER I. LINEAR SPACE

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pp. 1-44

In the euclidean plane E, we choose a fixed point 0 as the origin, and consider the set X of arrows or vectors in E with the common initial point 0 . A vector a in E with initial point 0 and endpoint A is by definition the ordered pair (O, A ) of points. The vector a = ( 0 , A) can be regarded as a graphical representation of a force acting at the ...

CHAPTER II. LINEAR TRANSFORMATIONS

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pp. 45-95

At the beginning of the last chapter, we gave a brief description of abstract algebra as the mathematical theory of algebraic systems and, in particular, linear algebra as the mathematical theory of linear spaces. These descriptions are incomplete, for we naturally want to find relations among the algebraic systems in question. In other ...

CHAPTER III. AFFINE GEOMETRY

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pp. 96-117

To define the basic notions of geometry, we can follow the so called synthetic approach by postulating geometric objects (e.g. points, lines and planes) and geometric relations (e.g. incidence and betweenness) as primitive undefined concepts and proceed to build up the geometry from a number of axioms which are postulated to ...

CHAPTER IV. PROJECTIVE GEOMETRY

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pp. 118-154

In order to have a concise theory without all these awkward exceptions, we can - and this is a crucial step towards projective geometry - extend the plane A (and similarly the plane A') by the adjunction of a set of new points called points at infinity. More precisely, we understand by a point at infinity of A the direction ...

CHAPTER V. MATRICES

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pp. 155-195

This therefore suggests the notion of a matrix as a doubly indexed family of scalars. Matrices are one of the most important tools in the study of linear transformations on finite-dimensional linear spaces. However, we need not overestimate their importance in the theory of linear algebra since the matrices play for the linear transformations ...

CHAPTER VI. MULTILINEAR FORMS

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pp. 196-229

Linear transformations studied in Chapter II are, by definition, vector-valued functions of one vector variable satisfying a certain algebraic requirement called linearity. When we try to impose similar conditions on vector-valued functions of two (or more) vector variables, two different points of view are open to us. To be ...

CHAPTER VII. EIGENVALUES

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pp. 230-266

Given a single endomorphism o of a finite-dimensional linear space X, it is desirable to have a base of X relative to which the matrix of o takes up a form as simple as possible. We shall see in this chapter that some endomorphisms can be represented (relative to certain bases) by matrices of diagonal form; while for every ...

CHAPTER VIII. INNER PRODUCT SPACES

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pp. 267-305

We began in Chapter I by considering certain properties of vectors in the ordinary plane. Then we used the set V² of all such vectors together with the usual addition and multiplication as a prototype linear space to define general linear spaces. So far we have entirely neglected the metric aspect of the linear space V² ; this means that ...

INDEX

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pp. 306-309