## Vectors

A vector is an array of numbers, x = (x1,x2, . . . ,xn), and may be interpreted in a number of ways: as the values of a set of variables, as a coordinate position in Rn,

as a direction (move xi units in direction i, for i = 1,. . . ,n), and so on. Depending on context, different interpretations are appropriate. In what follows, (Sections 9.2

and 9.3), some of the key summarizing features of a vector are considered: the

length of a vector, the distance between vectors and the angle between vectors.

As mentioned, from a geometric perspective, vectors can be interpreted as hav-

ing direction – in subsequent chapters on optimization, direction of improvement

of a function is central to the characterization of an optimal choice. As it turns

out, this direction is represented by a vector of partial derivatives of the func-

tion, the gradient vector. Vectors implicitly define hyperplanes and halfspaces,

key concepts in economic theory – Section 9.4 discusses these concepts. Finally,

Section 9.5 describes Farkas’ lemma, an old and important result in optimization

theory.