Vector Space Examples
Various examples of vector spaces are:
Real Numbers (ℝ): Set of all real numbers forms a vector space under standard addition and scalar multiplication. For example, any two real numbers can be added together (resulting in another real number), and any real number can be multiplied by a scalar (another real number) to give another real number.
Euclidean Space (ℝn): This is the classic n-dimensional vector space where vectors are represented as n-tuples of real numbers. For example, in ℝ3 (3-dimensional Euclidean space), vectors could be represented as (x, y, z), where x, y, and z are real numbers.
Polynomials: Set of all polynomials with coefficients from a field (like ℝ or ℂ) forms a vector space. For example, the set of all quadratic polynomials ax2 + bx + c, where a, b, and c are real numbers, is a vector space under polynomial addition and scalar multiplication.
Matrices: Set of all matrices of a fixed size (e.g., m x n matrices) with entries from a field forms a vector space. Matrices can be added together element-wise, and scalar multiplication involves multiplying each element of the matrix by a scalar.
What is Difference between Vector and Vector Space?
A vector is a mathematical object that has both magnitude and direction, while a vector space is a mathematical structure consisting of a set of vectors along with operations of addition and scalar multiplication, satisfying specific properties. Vectors are elements of vector spaces, providing the algebraic framework for studying linear relationships and operations.
Is Zero a Vector Space?
A set containing only the zero vector is called a vector space, it is also called a Zero vector Space(Trivial Vector Space). This vector satisfies all the axion of vector space and hence is called vector space.
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