Vector Space Definition

Scalars are often taken to be real numbers but there are also vector spaces with scalar multiplication by complex numbers rational numbers or generally any field.

Vector space definition. Denition of a vector space. But there are few cases of scalar multiplication by rational numbers complex numbers etc. In linear algebra the quotient of a vector space v by a subspace n is a vector space obtained by collapsing n to zero.

A vector space is a set v on which two operations and are defined called vector addition and scalar multiplication. Each element in a vector space is a list of objects that has a specific length which we call vectors. A vector space is a set that is closed under finite vector addition and scalar multiplication.

Conversely s is called a spanning set of w and we say that s spans w. Definition 1 7 a one element vector space is a trivial space. For all vectors u and v in v u v v u.

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. Definition of vector space. W is referred to as the subspace spanned by s or by the vectors in s.

A vector space or a linear space is a group of objects called vectors added collectively and multiplied scaled by numbers called scalars. 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. The operation vector addition must satisfy the following conditions.

This way the theorems start with the phrase let v be a vector space instead of the vague rambling phrase above. Scalars are usually considered to be real numbers. A vector space is a space in which the elements are sets of numbers themselves.