vector space zero ring
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.
Geometry: In a plane figure, a point which is a common end-point for two or more lines or curves.
Graph Theory: One of two kinds of entities in a graph.
The set of vertices of some graph. For a graph G, the vertex set of G is denoted by V(G), or, if there is no ambiguity as to the graph in question, simply by V.
In set theory, a collection is well-founded if every subcollection has a least member under the membership relation. For example, the set of natural numbers is well-founded. In ZFC, the foundation axiom asserts this property of all sets. A set which is not well-founded is sometimes called a hyperset.
An element of the set consisting of the number 0 together with the counting numbers, 1, 2, 3, etc.; i.e., the set N of natural numbers with 0 included. Sometimes the term “whole” is meant to refer to negative integers also; the intended meaning should be clear from the context.
The ring with only one element, its additive identity.