Vectors are an intuitive concept that we get acquainted with in highschool mathematics. In ordinary two- and three-dimensional geometry, we deal with points on the plane or in the space. Such points are associated with coordinates in a Cartesian reference system. Coordinates may be depicted as vectors, as shown in Fig. 3.4; in physics, vectors are associated with a direction (e.g., of motion) or an intensity (e.g., of a force). In this book, we are interested in vectors as tuples of numbers. The vectors of Fig. 3.4 can be represented numerically as a pair and a triple of numbers, respectively:
More generally, we define a vector in an n-dimensional space as a tuple of numbers:
As a notational aid, we will use boldface lowercase letters (e.g., a, b, x) to refer to vectors; components (or elements) of a vector will be referred to by lowercase letters with a subscript, such as y1 or vi. In this book, components are assumed to be real numbers; hence, an n-dimensional vector v belongs to the set 
of tuples of n real numbers. If n = 1, i.e., the vector boils down to a single number, we speak of a scalar element.
Note that usually we think of vectors as columns. Nothing forbids the use of a row vector, such as
In order to stick to a clear convention, we will always assume that vectors are column vectors; whenever we insist on using a row vector, we will use the transposition operator, denoted by superscript T, which essentially swaps rows and columns:
Sometimes, transposition is also denoted by v′.
In business applications, vectors can be used to represent several things, such as
- Groups of decision variables in a decision problem, such as produced amounts xi for items i = 1, …, n, in a production mix problem
- Successive observations of a single random variable X, as in a random sample (X1, X2, …, Xm)
- Attributes of an object, measured according to several criteria, in multivariate statistics
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