Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

APPENDIX A Vectors and Matrices

A.1Introduction

Data is frequently arranged inarrays, 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 avectorand a two-dimensional array is

called amatrix(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.2Vectors

By avector u, we mean a list of numbers, say,a 1 ,a 2 ,...,an. Such a vector is denoted by

u=(a 1 ,a 2 ,...,an)

The numbersaiare called thecomponentsorentriesofu. If all theai=0, thenuis called thezero vector.Two

such vectors,uandv, areequal, writtenu=v, if they have the same number of componentsandcorresponding

components are equal.

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 vectorsuandvwith the same number of components, say

u=(a 1 ,a 2 ,...,an) and v=(b 1 ,b 2 ,...,bn)

409

Copyright © 2007, 1997, 1976 by The McGraw-Hill Companies, Inc. Click here for terms of use.