2.2.1 Cartesian Product

2.2.1  Cartesian Products

An ordered pair is a set containing two elements with an ordering that identifies one element as being “first” and the other element as being “second.” The set {yellow, red} equals the set {red, yellow}, but the ordered pair (yellow, red) does not equal the ordered pair (red, yellow). Similarly, an n-tuple is an ordered set containing n elements. As in our example, we indicate ordered pairs and ordered n-tuples with parentheses ( ) to distinguish them from sets, which we indicate with brackets { }.1

Let A and B be sets. The Cartesian product A × B is the set of ordered pairs (a, b) where a A and b B. More generally, let A1, A2, … , An be n sets. The Cartesian product A1 × A2 × … × An is the set of n-tuples (a1, a2, … , an} where ai Ai. Shorthand notation for the Cartesian product of a set with itself is A2 = A × A, or more generally, An = A × A × …  × A. In this manner, 2 represents 2-dimensional space and n represents n-dimensional space.

2.2.2 Vectors

We assume familiarity with vectors, which are elements of—n-tuples or “points” in—n. We distinguish between column vectors and row vectors. Unless stated otherwise, all vectors are assumed to be column vectors. There are two different notations used for vectors. One is matrix notation, which indicates column vectors x and row vectors y as


More compact is n-tuple notation, which we have already introduced. This indicates the same column and row vectors as


A distinguishing characteristic is that n-tuple notation employs commas whereas matrix notation does not. With either notation, we may use a prime ′ to indicate transposition. We denote a zero vector (0, 0, … , 0) as 0.

The norm or length of a vector x is defined as


In one dimension, length reduces to absolute value, which we denote |x|.

2.2.3 Matrices

We assume familiarity with matrices. We indicate a zero matrix as 0 and an identity matrix as I. As with vectors, we indicate matrix transposition with a prime ′, so h′ is the transpose of h. We denote a matrix inverse as h–1 and a determinant as |h|. Components of matrices are assumed to be real numbers unless stated otherwise. Matrix addition and matrix multiplication require that matrices have compatible dimensions. Such compatibility is assumed whenever we apply these operations.