Introduction to Vector Spaces
December 01, 2024 | S.A.D.Archana Methviru
Introduction to Vector Spaces
A mathematical entity with both magnitude and direction is called a vector. A vector can be represented geometrically as a directed line segment, with the arrow indicating the direction of the vector and the segment's length representing its magnitude. More precisely, the direction is from the vector's tail to its head. Vectors are components of a vector space, a mathematical structure made up of sets of numbers, in a more abstract sense. An n-tuple is an element in this space that has a defined length for each component. These vectors are essential for describing linear functions in a variety of applications, especially in linear algebra(“Vector Space,” 2021).
The two operations that can be carried out inside a vector space are vector addition and scalar multiplication. While scalar multiplication entails scaling a vector by a numerical factor (a scalar), it involves increasing a vector's magnitude without changing its direction. Vector addition creates a third vector by combining two vectors. The main characteristic of these operations is that they always produce outcomes that remain in the vector space, hence maintaining the vectors' original nature(Vector Space, 2023). The foundation of vector spaces, represented by symbols like 𝑈, 𝑉, and 𝑊, is this combination of vectors (as parts of a group) and scalars (as elements of a field). The study of linear algebra and its many applications in science and engineering require a fundamental understanding of these operations and the characteristics of vector spaces.
Apart from the basic vector operations, vector addition and scalar multiplication in a given space V have to follow certain conditions, called axioms, in order for the space to be considered a vector space. The general characteristics of vectors in field F are described by these axioms. The vector space is called a real vector space when it is over the set of real numbers R. The elements of V can be regarded as vectors, and the linear algebraic theorems will apply to them if the vector addition and scalar multiplication procedures follow the vector space axioms.
If we consider a set F with two binary operations, addition and multiplication, where the sum and product of two elements a and b in F are represented as a + b and a · b respectively, and these operations adhere to the following rules, then F is referred to as the field of a vector space.
Addition is commutative, i.e. a+b = b+a.
Addition is associative, i.e. a+(b+c) = (a+b)+c.
Multiplication is commutative, i.e. ab = ba.
Multiplication is associative, i.e. a(bc) = (ab)c.
Multiplication is distributive, i.e. a(b+c) = ab+ac.
F contains additive as well as multiplicative identity as 0 and 1 respectively, with the property as a+0=a, and 1a=a, for every a in F.
F contains an additive inverse of a that is -a, such that a+(-a)=0.
For all non-zero elements in F, it contains a multiplicative inverse that is a-1, such that aa−1=1.
If a non-empty subset W of a vector space V itself constitutes a vector space under the addition and scalar multiplication operations specified in V, then W is referred to as a subspace of V. In general, one needs to prove each of the 10 vector space axioms in order to show that a set W with these operations is a vector space. Nevertheless, W inherits some axioms from V if W is a subset of a bigger set V that is already known to be a vector space; therefore, W does not require independent verification of these axioms. Thus, if W meets the requirements derived from V, it can be regarded as a subspace of V(Wawro et al., 2011).
An expression created by multiplying each term in a set by a constant and adding the results is known as a linear combination. In the setting of vector spaces, a linear combination is represented by the expression 𝑎1𝑣1 + 𝑎2𝑣2 + ⋯ + 𝑎𝑛𝑣𝑛, given vectors 𝑣1, 𝑣2, …, 𝑣𝑛 in a vector space V and scalars 𝑎1, 𝑎2, …, 𝑎𝑛 from a field F. Since it explains how vectors can be scaled and added to form new vectors inside the same space, this idea is essential to the study of linear algebra. Let S be a nonempty subset of V and let V be a vector space over the field of real numbers R. If there are vectors 𝑦1,𝑦2,…,𝑦𝑛 in S and scalars 𝛼1,𝛼2,…,𝛼𝑛 such that 𝑥 = 𝛼1𝑦1 + 𝛼2𝑦2 + ⋯ + 𝛼𝑛𝑦𝑛. then a vector 𝑥 ∈ 𝑉 is considered a linear combination of the vectors in S. This straightforward method of adding and scaling vectors is a useful tool for comprehending and working with vector spaces(Trybulec, 2003).
A set can be expressed by a collection of vectors if each vector in the set can be represented as a linear combination of the vectors in the collection. All conceivable linear combinations of a set S of vectors are represented by the set known as span(S). In essence, span(S) is a representation of every vector that may be created by adding and scaling the vectors in S. Any vector in a vector space may be built from a set of vectors if they span the vector space, which makes them the essential building blocks of the space. This idea is essential to comprehending the structure of vector spaces since it tells us if a given set of vectors may serve as the space's basis.
The ideas of linear dependency and independence are essential to comprehending the dimension and structure of a space in vector space theory. If there is a nontrivial linear combination of these vectors that equals the zero vector, then the collection of vectors is said to be linearly dependent. This indicates that not all scalars are zero in order for the combination 𝑎1𝑣1 + 𝑎2𝑣2 + ⋯ + 𝑎𝑘𝑣𝑘 to equal 0. In these situations, the set's vectors can all be expressed as linear combinations of one another. On the other hand, if the trivial linear combination—in which every coefficient is zero—is the only one that yields a zero vector, then the set of vectors is said to be linearly independent.
This suggests that there isn't a single vector in the collection that can be expressed as a linear combination of all the others. Thus, if the equation 𝑎1𝑣1 + 𝑎2𝑣2 + ⋯ + 𝑎𝑛𝑣𝑛 = 0 can only be met when 𝑎𝑖 = 0 for all 𝑖, then a sequence of vectors 𝑣1 , 𝑣2, …, 𝑣𝑛 is linearly independent. Since the greatest number of linearly independent vectors in a vector space determines its dimension, these ideas are essential to understanding the definition of vector space dimensions(“Linear Independence,” 2024).
The basis of a vector space is a set of vectors that are both linearly independent and span the entire space. Consider a subspace V of 𝑅𝑛. A set 𝐵 = {𝑣1, 𝑣2, …, 𝑣𝑟} of vectors from V is defined as a basis for V if B meets two key criteria: it must be linearly independent and it must span V. If either of these conditions is not fulfilled, then the set is not a basis for V. When a set of vectors spans V, it means there are enough vectors in the set to express every vector in V as a linear combination of those vectors.
A set is said to be linearly independent if its vector count is small enough to preclude any one of its vectors from being expressed as a linear combination of its other vectors. Naturally, a basis has the ideal quantity of vectors: enough to cover the whole space without duplication, guaranteeing that no vector depends on any other vector(Basis of a Vector Space | Definition & Examples - Lesson, 2023).
