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Vectors and Matrices

A.1 INTRODUCTION

Data is frequently arranged in arrays, that is, sets whose elements are indexed by one or more subscripts.

If the data consists of numbers, then a one-dimensional array is called a vector and a two-dimensional array is

called a matrix (where the dimension denotes the number of subscripts). This appendix investigates these vectors

and matrices, and certain algebraic operations involving them. In this context, the numbers themselves are called

scalars.

A.2 VECTORS

By a vector u, we mean a list of numbers, say, a1, a2, . . . , an. Such a vector is denoted by

u = (a1, a2, . . . , an)

The numbers ai are called the components or entries of u. If all the ai = 0, then u is called the zero vector. Two

such vectors, u and v, are equal, written u = v, if they have the same number of components and corresponding

EXAMPLE A.1

(a) The following are vectors where the first two have two components and the last two have three components:

(3,?4), (6, 8), (0, 0, 0), (2, 3, 4)

The third vector is the zero vector with three components.

(b) Although the vectors (1, 2, 3) and (2, 3, 1) contain the same numbers, they are not equal since corresponding

components are not equal.

Vector Operations

Consider two arbitrary vectors u and v with the same number of components, say

u = (a1, a2, . . . , an) and v = (b1, b2, . . . , bn)

The sum of u and v, written u + v, is the vector obtained by adding corresponding components from u and v;

that is,

u + v = (a1 + b1, a2 + b2, . . . , an + bn)

The scalar product or, simply, product, of a scalar k and the vector u, written ku, is the vector obtained by