Vector Space Definition

A vector space is a space in which the elements are sets of numbers themselves.

Vector space definition. A vector space also called a linear space is a collection of objects called vectors which may be added together and multiplied scaled by numbers called scalars. Scalars are usually considered to be real numbers. Given a vector space v over a field k the span of a set s of vectors not necessarily infinite is defined to be the intersection w of all subspaces of v that contain s. The operation vector addition must satisfy the following conditions.

Conversely s is called a spanning set of w and we say that s spans w. A vector space is a set v on which two operations and are defined called vector addition and scalar multiplication. Definition 1 7 a one element vector space is a trivial space. The space obtained is called a quotient space and is denoted v n read v mod n or v by n.

This way the theorems start with the phrase let v be a vector space instead of the vague rambling phrase above. Definition of vector space. W is referred to as the subspace spanned by s or by the vectors in s. Denition of a vector space.

A set of vectors along with operations of addition and multiplication such that the set is a commutative group under addition it includes a multiplicative inverse and multiplication by scalars is both associative and distributive examples of vector space in a sentence. For all vectors u and v in v u v v u. In linear algebra the quotient of a vector space v by a subspace n is a vector space obtained by collapsing n to zero. The basic example is dimensional euclidean space where every element is represented by a list of real numbers scalars are real numbers addition is componentwise and scalar multiplication is multiplication on each term separately.

A vector space is a set that is closed under finite vector addition and scalar multiplication. But mathematicians like to be concise so they invented the term vector space to mean any type of mathematical object that can be multiplied by numbers and added together. But there are few cases of scalar multiplication by rational numbers complex numbers etc. If u and v are any vectors in v then the sum u v belongs to v.