# Linear Algebra and Geometry

Publication Year: 1974

Published by: Hong Kong University Press, HKU

#### Cover

#### Title Page

#### Copyright

#### PREFACE

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 ...

#### CHAPTER I. LINEAR SPACE

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

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

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

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

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

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

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

We began in Chapter I by considering certain properties of vectors
in the ordinary plane. Then we used the set *V*^{2} 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 *V*^{2} ; this means that ...

E-ISBN-13: 9789882202078

Print-ISBN-13: 9780856561115

Page Count: 318

Publication Year: 1974

OCLC Number: 652681936

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