# Vectors, Matrices and Geometry

K.T. Leung
S.N. Suen
Pages: 356
https://www.jstor.org/stable/j.ctt2jc242

1. Front Matter
(pp. i-iv)
(pp. v-vi)
3. PREFACE
(pp. vii-viii)
K.T. Leung
4. Chapter One VECTORS AND GEOMETRY IN THE PLANE
(pp. 1-38)

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 passing through $P$ and $Q$ consists of points $X$ whose coordinates $(x,\ y)$ satisfy the polynomial equation $(y\ -\ b)(c\ -\ a)\ =\ (x\ -\ a)(d\ -\ b)$ and the slope of the line is given by the algebraic expression $(d\ -\ b)/(c\ -\ a)$ in the coordinates of $P$ and $Q$ . We note that here algebraic operations are not carried out on...

5. Chapter Two VECTORS AND GEOMETRY IN SPACE
(pp. 39-88)

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 ${{\bf R}^{3}}$ 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 small selection of classical theorems.

The cartesian coordinates of a point in the plane are given in relation to a pair of mutually perpendicular axes; those of...

6. Chapter Three CONIC SECTIONS
(pp. 89-166)

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 attacking the Delian problem of duplication of a cube, found himself facing the task of constructing two mean proportionals $x$ and $y$ between two given line segments of length $a$ and $b$ : $$a\ :\ x\ =\ x\ :\ y\ =\ y\ :\ b\ .$$

From...

(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 circular cone in space. In Chapter Three we have seen that there are only three types of conics. A quadratic equation in three variables defines a surface in...

8. Chapter Five HIGHER DIMENSIONAL VECTOR SPACES
(pp. 207-242)

In the first two chapters, we have learnt the language and techniques of linear algebra of vectors in ${{\bf R}^{\bf 2}}$ and ${{\bf R}^{\bf 3}}$ . Instead of moving up one dimension from ${{\bf R}^{\bf 3}}$ to ${{\bf R}^{\bf 4}}$ , we shall study the general $n$ -dimensional vector space ${{\bf R}^{\bf n}}$ 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 and of subspace of ${{\bf R}^{\bf n}}$ . In order to study the notion of dimension properly, we find it necessary to introduce matrices and elementary transformations on...

9. Chapter Six MATRIX AND DETERMINANT
(pp. 243-280)

Matrices are introduced in the last chapter as a systematic way of presenting the components of $m$ vectors of ${{\bf R}^{\bf n}}$ 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 on their own right. Sums and products of matrices as well as their properties are studied in the first part of this chapter. A particularly interesting and useful result is the interpretation of elementary transformations on...

10. Chapter Seven LINEAR EQUATIONS
(pp. 281-314)

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 transformations.

A linear equation in $n$ unknowns ${{x}_{1}},{{x}_{2}},\cdots,{{x}_{n}}$ is an expression of the form $${{a}_{1}}{{x}_{1}} + {{a}_{2}}{{x}_{2}} + \cdots + {{a}_{n}}{{x}_{n}} = b {\caption {(1)}}$$

where the coefficients ${{a}_{1}},{{a}_{2}},\cdots ,{{a}_{n}}$ and the constant term $b$ are all fixed real numbers such that at least...