BROWSE
ALPHABETICALLY
LEVEL:
Elementary
Advanced
Both
INCLUDE TOPICS:
Basic Math
Algebra
Analysis
Biography
Calculus
Comp Sci
Discrete
Economics
Foundations
Geometry
Graph Thry
History
Number Thry
Phys Sci
Statistics
Topology
Trigonometry
|
|
unit square – well-ordering principle
unit square
The set of points of the Cartesian plane with domain and range values in the unit interval, that is the square region with vertices (0, 0), (0, 1), (1, 0), and (1, 1), including its boundary.
upper bound
An upper bound of a set with an order relation (such as “ < ”) defined on it is an element which is greater than or equal to every element in the set.
Cf. least upper bound.
vector
A quantity having two components; a magnitude component and a direction component. In n-dimensional Euclidean space, a vector is representable by an ordered tuple (a1, a2, a3, ... an) whose elements are called the components of the vector. In this case the magnitude of the vector is given by the square root of the sum of the squares of the components of the vector.
vector product
The vector product (also called cross product) of two vectors u and v, denoted u × v and called “u cross v,” is a vector w whose magnitude (length) is the product of the magnitudes of u and v and the sine of the angle between them, and which points in a direction perpendicular to the plane containing u and v so as to form a right-handed system, as in the figure.
Note that the directedness of the vector product implies that it is not commutative.
Cf. scalar product.
vector space
A structure consisting of two kinds of elements called scalars and vectors, with operations of addition of pairs of scalars or pairs of vectors, and multiplication of pairs of scalars or a scalar and a vector. The vectors form an Abelian group under addition, and the scalars form a field under their operations, and the vector space is said to be over that field.
If the scalar field is the real numbers or the complex numbers and the vectors are in n-dimensional real or complex space, then the space is called an n-dimensional real or complex vector space accordingly. Multiplication of vectors by scalars is associative with scalar multiplication, and distributive over both scalar addition and vector addition. Symbolically, for scalars a, b, and vectors u, v,
Vector spaces are usually denoted by V, and it is conventional to write the scalar on the left of a scalar multiplication. When there is any possibility of confusion, the vectors of a vector space are usually specially marked, either by drawing a (right pointing) arrow over them or by writing them in bold face.
Cf. module.
well-ordered
A set S with a linear order is called well-ordered if every non-empty subset T of S has a least element under the ordering relation.
Cf. well-ordering principle.
well-ordering principle
The assertion that every set can be well-ordered. Equivalent to the Axiom of Choice.
|
|
 
|