Vector Space Axioms
Ten axioms can define vector space. Let x, y, & z be the elements of the vector space V and a & b be the elements of the field F.
1. Closed Under Addition
For every element x and y in V, x + y is also in V.
2. Closed Under Scalar Multiplication
For every element x in V and scalar a in F, ax is in V.
3. Commutativity of Addition
For every element x and y in V, x + y = y + x.
4. Associativity of Addition
For every element x, y, and z in V, (x + y) + z = x + (y + z).
5. Existence of the Additive Identity
There exists an element in V which is denoted as 0 such that x + 0 = x, for all x in V.
6. Existence of the Additive Inverse
For every element x in V, there exists another element in V that we can call -x such that x + (-x) = 0.
7. Existence of the Multiplicative Identity
There exists an element in F notated as 1 so that for all x in V, 1x = x.
8. Associativity of Scalar Multiplication
For every element x in V, and for each pair of elements a and b in F, (ab)x = a(bx).
9. Distribution of Elements to Scalars
For every element a in F and every pair of elements x and y in V, a(x + y) = ax + ay.
10. Distribution of Scalars to Elements
For every element x in V, and every pair of elements a and b in F, (a + b)x = ax + bx
Vector Space- Definition, Axioms, Properties and Examples
A vector space is a group of objects called vectors, added collectively and multiplied by numbers, called scalars. Scalars are usually considered to be real numbers. But there are few cases of scalar multiplication by rational numbers, complex numbers, etc. with vector spaces.
In this article, we have covered Vector Space Definition, Axions, Properties and others in detail.
Table of Content
- What is Vector Space?
- Vector Space Axioms
- Vector Space Examples
- Dimension of a Vector Space
- Vector Addition and Scalar Multiplication
- Vector Space Properties
- Subspaces
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