Numerical Methods and Optimization

8 Numerical Methods and Optimization: An Introduction

Definition 1.5 (Real Vector) Arealn-dimensional real vector is an

ordered set of n real numbers {x

1

,x

2

,...,x

n

} and is usually written in

the form

x =

⎡

⎢

⎢

⎢

⎣

x

1

x

2

.

.

.

x

n

⎤

⎥

⎥

⎥

⎦

(column vector) or

x =[x

1

,x

2

,...,x

n

]

(row vector). We will also write

x =[x

i

]

n

i=1

.

The numbers x

1

,x

2

,...,x

n

are called the components of x.

For the sake of clarity, unless otherwise specified, by a vector we will mean a

column vector. For a vector x, the corresponding row vector will be denoted

by x

T

, which represents the transpose of x (see Definition 1.25 at page 14).

Definition 1.6 (Real Coordinate Space) The set of all n-dimensional

ve