# Linear Algebra and Geometry

KAM-TIM LEUNG
Pages: 318
https://www.jstor.org/stable/j.ctt2jc6mm

1. Front Matter
(pp. i-iv)
2. PREFACE
(pp. v-vi)
K. T. Leung
(pp. vii-ix)
4. [Illustration]
(pp. x-x)
5. CHAPTER I LINEAR SPACE
(pp. 1-44)

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

6. CHAPTER II LINEAR TRANSFORMATIONS
(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 words, we also have to study mappings between algebraic systems compatible with the algebraic structure in question.

A linear space consists, by definition, of two parts: namely a non-empty set and an algebraic structure on this set. It is therefore natural to compare one linear space with...

7. CHAPTER III AFFINE GEOMETRY
(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 govern and regulate these primitive concepts. No doubt this approach is most satisfactory from the aesthetic as well as from the logical point of view. However it will take quite a few axioms to develop the subject beyond a few trivial theorems, and the choice of a system of...

8. CHAPTER IV PROJECTIVE GEOMETRY
(pp. 118-154)

Let $A$ and $A^{\prime}$ be two distinct planes in the ordinary space, and let $O$ be a point which is neither on $A$ nor on $A^{\prime}$ . The central projection $p$ of $A$ into $A^{\prime}$ with respect to the centre of projection $O$ is defined as follows: for any $Q$ on $A$ we set $p(Q)\hspace{3pt}=\hspace{3pt}Q^{\prime}$ if the points $Q$ , $Q^{\prime}$ and $O$ are collinear (i.e. on a straight line). If $A$ and $A^{\prime}$ are parallel planes, then $p$ is an affinity of the 2-dimensional affine space $A$ onto the 2-dimensional affine space $A^{\prime}$ . In particular, $p$ is a...

9. CHAPTER V MATRICES
(pp. 155-195)

In Examples 5.8, we gave some effective methods of constructing linear transformations. Among others, we saw that for any finite-dimensional linear spaces $X$ and $Y$ over Λ with bases $\left(x_{1},\hspace{3pt}...,\hspace{3pt}x_{m}\right)$ and $\left(y_{1},\hspace{3pt}...,\hspace{3pt}y_{n}\right)$ respectively a unique linear transformation $\phi:\hspace{5pt}X\hspace{5pt}\rightarrow\hspace{5pt}Y$ is determined by a family $\left(a_{ij}\right)_{i=1,\hspace{3pt}...,\hspace{3pt}m;\hspace{3pt}j=1,\hspace{3pt}...,\hspace{3pt}n}$ of scalars in such a way that

$\phi\left(x_{i}\right)\hspace{3pt}=\hspace{3pt}a_{i1}y_{1}\hspace{5pt}+\hspace{5pt}...\hspace{5pt}+\hspace{3pt}a_{in}y_{n}\hspace{5pt}\text{for}\hspace{5pt}i\hspace{3pt}=\hspace{3pt}1,\hspace{3pt}...,\hspace{3pt}m.$

Conversely, let $\psi:\hspace{5pt}X\hspace{3pt}\rightarrow\hspace{3pt}Y$ be an arbitrary linear transformation. In writing each $\psi\left(x_{i}\right)$ as a linear combination of the base vectors $y_{j}$ of $Y$ , i.e.,

$\psi\left(x_{i}\right)\hspace{3pt}=\hspace{3pt}\beta_{i1}y_{1}\hspace{3pt}+\hspace{3pt}...\hspace{3pt}+\hspace{3pt}\beta_{in}y_{n}\hspace{3pt}\text{for}\hspace{3pt}i\hspace{3pt}=\hspace{3pt}1,\hspace{3pt}...,\hspace{5pt}m,$

we obtain a family $\left(\beta_{ij}\right)_{i=1,\hspace{2pt}\cdot\hspace{2pt}\cdot\hspace{2pt}\cdot\hspace{2pt},m;\hspace{3pt}j\hspace{3pt}=\hspace{3pt}1,\hspace{2pt}\cdot\hspace{2pt}\cdot\hspace{2pt}\cdot\hspace{2pt},n}$ of scalars. Thus relative to the bases $\left(x_{1},\hspace{5pt}...,\hspace{5pt}x_{m}\right)$ and $\left(y_{1},\hspace{5pt}...,\hspace{5pt}y_{n}\right)$ of $X$ and $Y$ respectively, each linear...

10. CHAPTER VI MULTILINEAR FORMS
(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 more precise, let us consider a mapping $\phi:\hspace{3pt}X\hspace{3pt}\times\hspace{3pt}Y\hspace{3pt}\rightarrow\hspace{5pt}Z$ where $X$ , $Y$ and $Z$ are all linear spaces over the same Λ. Now the domain $X\hspace{5pt}\times\hspace{5pt}Y$ can be either (i) regarded as the cartesian product of linear spaces and thus as a linear space in its own right or (ii) taken...

11. CHAPTER VII EIGENVALUES
(pp. 230-266)

Given a single endomorphism σ of a finite-dimensional linear space $X$ , it is desirable to have a base of $X$ relative to which the matrix of σ 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 endomorphism of a complex linear space we can find a base relative to which the matrix of the endomorphism is of Jordan form. In § 18 we give a rudimentary study on polynomials needed in the sequel. Characteristic polynomials will...

12. CHAPTER VIII INNER PRODUCT SPACES
(pp. 267-305)

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 we have only studied the qualitative concept of linearity and have omitted from consideration the quantitative concepts of length and angle of a linear space. Now we shall fill this gap in the present chapter. We use the 2-dimensional arithmetical...

13. INDEX
(pp. 306-309)