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.

What is Vector Space?

A space in mathematics comprised of vectors, that follow the associative and commutative law of addition of vectors and the associative and distributive process of multiplication of vectors by scalars is called vector space. In vector space, it consists of a set of V (elements of V are called vectors), a field F (elements of F are scalars) and the two arithmetic operations

Vector Addition: It is an operation that takes two vectors u, v ∈ V, and it produces the third vector u + v ∈ V

Scalar Multiplication: It is an operation that takes a scalar c ∈ F and a vector v ∈ V and produces a new vector uv ∈ V.

Vector Space Definition

A space comprised of vectors, collectively with the associative and commutative law of addition of vectors and also the associative and distributive process of multiplication of vectors by scalars is called vector space.

• Vector Addition is an operation that takes two vectors u, v ∈ V, and it produces the third vector u + v ∈ V

• Scalar Multiplication is an operation that takes a scalar c ∈ F and a vector v ∈ V and it produces a new vector uv ∈ V

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 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.

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, …, v_r} 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 Addition and Scalar Multiplication

Vector addition and scaler multiplication are two main concept in vector space that are explained below:

Vector Addition

When you add two vectors, you add their corresponding components. For example, if you have two vectors v = ⟨v₁, v₂, v₃⟩and w = ⟨w₁, w₂, w₃⟩ their sum v+ wv+ w is ⟨v₁+w₁, v₂+w₂, v₃+w₃⟩. Geometrically, vector addition represents the process of moving one vector's endpoint to the other vector's endpoint, forming a new vector from the initial point of the first vector to the final point of the second vector.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar. For example, if you have a vector ⟨v = v₁, v₂, v₃⟩ and a scalar k, then the scalar multiple kvis ⟨kv₁, kv₂, kv₃⟩. Geometrically, scalar multiplication stretches or compresses the vector without changing its direction, depending on whether the scalar is greater than 1 or between 0 and 1

Linear Combinations and Span

Let v₁, v₂,…, v_r be vectors in Rⁿ . A linear combination of these vectors is any expression of the form

k₁v₁ + k₂v₂ + ......... + k_rV_r

where the coefficients k₁, k₂,…, k_r are scalars.

Vector Space Properties

Some important properties of vector space are:

• Closure under Addition: Sum of any two vectors in the vector space is also a vector in the vector space.

• Closure under Scalar Multiplication: Multiplying any vector in the vector space by a scalar yields another vector in the vector space.

• Associativity of Addition: Vector addition is associative, meaning (u + v)+ w = u + (v + w) for all vectors u, v, and w in the vector space.

• Commutativity of Addition: Vector addition is commutative, meaning u + v = v + u for all vectors u and v in the vector space.

• Existence of Additive Identity: There exists a vector, denoted by 0 or 0, called the zero vector, such that u + 0 = u for all vectors u in the vector space.

• Existence of Additive Inverse: For every vector u in the vector space, there exists a vector -u such that u + (-u) = 0.

• Distributive Properties: Scalar multiplication distributes over vector addition, meaning α(u + v) = αu + αv and (α+β)u = αu + βu for all scalars α and β, and vectors u and v in the vector space.

• Multiplicative Identity: Scalar 1 acts as the multiplicative identity, meaning 1⋅u = u for all vectors u in the vector space.

Subspaces

A subset W of a vector space V is called a subspace of V if W is itself a vector space under the addition and scalar multiplication defined on V.

Subspaces are subsets of a vector space that themselves form vector spaces. Operations of vector addition and scalar multiplication from the larger vector space are applicable to vector space. Subspaces satisfies all axion/properties of vector space.

• Contain the zero vector
• Is closed under addition
• Is closed under scalar multiplication

They can be lower-dimensional spaces within the larger vector space and can provide insights into the structure and properties of the vector space as a whole.

Difference Between Vector Space and Euclidean Space

Applications of Vector Spaces

When an object is made up of multiple components it is often useful to represent the object as a vector, with one entry per component. The examples discussed in this section involve molecules, which are made up of atoms, and text documents, which are made up of words.

In some cases equations involving the objects give rise to vector equations. In other examples there are reasons to perform operations on the vectors using matrix algebra. Vector Spaces is also used in Machine Learning and its various other uses are:

• Data Representation: In many machine learning algorithms, data is represented as vectors. For example, images can be represented as vectors of pixel values, text documents can be represented as vectors of word counts or embeddings, and numerical data can be directly represented as vectors.

• Feature Vectors: Feature engineering involves creating meaningful representations of data. These representations are often in the form of feature vectors, where each feature corresponds to a dimension in the vector space. Feature vectors are used as input to machine learning models.

• Vector Operations: Vector operations such as addition, subtraction, dot products, and vector norms are commonly used in machine learning algorithms. For example, in clustering algorithms like k-means, vector addition and subtraction are used to calculate centroids.

• Linear Algebra in Models: Many machine learning models are based on linear algebra operations. For example, linear regression involves finding a line that best fits a set of data points, which can be formulated as a linear algebra problem involving vectors and matrices.