Dimension of a Vector Space
Number of vectors in a basis for V is called the dimension of V.
For example, the dimension of Rn is n. The dimension of the vector space of polynomials in x with real coefficients having degree at most two is 3.
Basis of Vector Space
Let V be a subspace of Rn for some n. A collection B = {v1, v2, …, vr} of vectors from V is said to be a basis for V if B is linearly independent and spans V. If either one of these criterial is not satisfied, then the collection is not a basis for V.
If a collection of vectors spans V, then it contains enough vectors so that every vector in V can be written as a linear combination of those in the collection. If the collection is linearly independent, then it doesn’t contain so many vectors that some become dependent on the others.
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
Contact Us