Vector Spaces

What is a Vector Space?

A vector space over a field K is a non-empty set of vectors V equipped with two binary operations (vector addition and scalar multiplication) that adhere to certain properties.

It's also referred to as a linear space or a K-vector space.

Visually, a vector space is the collection of all vectors that originate from a single point, combined with the operations of vector addition and scalar multiplication of vectors.

Note. Intuitively, the elements of a vector space are vectors. However, in general, a vector space can also consist of other objects, such as a space of matrices or a space of polynomials.

What are the components of a Vector Space?

The components of a vector space are as follows:

• A field (K), the elements of which are called scalars.
• A non-empty set (V), the elements of which are called vectors.
• Two binary operations: vector addition and scalar multiplication, which satisfy all of the following properties:
  • Commutative property $$ \vec{v}+\vec{w}=\vec{w}+\vec{v} \ \ \ \ \ \ \vec{v}, \vec{w} \in V $$
  • Associative property $$ ( \vec{u}+\vec{v} ) +\vec{w} = \vec{u} + ( \vec{v}+\vec{w} ) \ \ \ \ \ \ \vec{v}, \vec{w}, \vec{u} \in V $$
  • Existence of a zero vector $$ \exists \ \vec{0} \ \in V \ \ | \ \ \vec{v}+\vec{0} = \vec{0}+\vec{v} = \vec{v} $$
  • Existence of an additive inverse $$ \forall \ \vec{v} \ \ \exists \ -\vec{v} \ \in V \ \ | \ \ \vec{v}+(-\vec{v}) = \vec{0} $$
  • Scalar multiplication $$ (a \cdot b) \cdot \vec{v} = a \cdot (b \cdot \vec{v} ) \ \ \ \ a,b \in K \ \ \ \vec{v} \in V $$

    Note. The multiplication dot signifies two different operations. The product a*b is a multiplication of two scalars in the field K, while (a*b)*v is the multiplication of a scalar (a*b) by a vector v.

  • Distributive property $$ (a + b) \cdot \vec{v} = a \cdot \vec{v} + b \cdot \vec{v} \ \ \ \ \ \ \ \ a,b \in K \ \ \ \vec{v} \in V $$ $$ a \cdot (\vec{v} + \vec{w}) = a \cdot \vec{v} + a \cdot \vec{w} \ \ \ \ \ \ \ \ a \in K \ \ \ \vec{v}, \vec{w} \in V $$
  • Identity element of scalar multiplication $$ 1 \cdot \vec{v} = \vec{v} \ \ \ \ \ \ \ \ \forall \vec{v} \in V $$

It should be emphasized that the properties of operations within the vector space V are similar to, yet distinct from, the properties of operations within the field K.

In the vector space V, operations are conducted between vectors or between vectors and scalars, whereas in the field K, operations occur between scalars.

Therefore, in a vector space, operations among scalars must satisfy the properties of the field K, and operations involving vectors must satisfy the properties of the vector space V.

For a more in-depth look at the properties of operations in a vector space.

What is a Field? In mathematics, a field is an algebraic structure consisting of a non-empty set K and two binary operations (addition and multiplication) denoted by + and *.

If the field K is the set of real numbers R, the space is called a real vector space. If the field is the set of complex numbers C, the space is termed a complex vector space.

An Example of a Vector Space

Example 1: The Vector Space R¹

The vector space R¹ is a one-dimensional space (n=1) over the field of real numbers K=R, where the operations of vector addition and scalar multiplication are defined.

Since n=1, the vector space is identical to the field of real numbers itself.

$$ R^1 = \{ (a_1) \} \ \ \ \ \ \ a_1 \in R $$

Every point on the line corresponds to a real number.

With a fixed origin (O), every point on the line represents a vector whose magnitude (modulus) is equal to a real number.

Note. In the vector space R¹, all vectors have the same direction, are oriented one way or the other, and have different lengths (moduli).

Here are some examples of vectors in R¹

$$ \vec{v_1} = 3 \ \ \ \ \vec{v_2} = 5 \ \ \ \ \vec{v_3} = -1 \ \ \ ... $$

The operations of vector addition and scalar multiplication in R¹ satisfy the properties of vector spaces.

Example 2: The Vector Space R²

When n=2, the vector space R² (i.e., RxR) corresponds to the Cartesian plane of real numbers.

$$ R^2 = \{ (a_1,a_2) \} \ \ \ \ \ \ a_1,a_2 \in R $$

Here, RxR is the Cartesian product of the set of real numbers.

In this case, each pair of real numbers (x,y) corresponds to a point in Euclidean geometry, that is, a two-dimensional vector.

Thus, the real vector space is composed of R² vectors.

Here are some examples of vectors consisting of n=2 elements in the field of real numbers R.

Note. Since the set of real numbers R is infinite, the real vector space R² is also infinite.

In the vector space R², two binary operations are defined: vector addition and scalar multiplication of a vector.

All operations adhere to the properties of operations in vector spaces.

This demonstrates that R² is a vector space over the field R.

The same reasoning can be applied to any other real vector space with Rⁿ numeric vectors.

Example 3: The Vector Space R³

If n=3, the vector space R³ (i.e., RxRxR) corresponds to the three-dimensional space.

$$ R^3 = \{ (a_1,a_2, a_3) \} \ \ \ \ \ \ a_1,a_2, a_3 \in R $$

In this case, each triplet of real numbers (x,y,z) corresponds to a point in space, that is, a three-dimensional vector where a₁=x, a₂=y, a₃=z.

Also, in the vector space R³, the two binary operations are defined: vector addition and scalar multiplication of a vector.

The operations comply with all the properties of vector spaces.

Example 4: The Vector Space R⁴

For n=4, the vector space R⁴ (i.e., RxRxRxR) is a four-dimensional space.

It is a space composed of vectors with four components (a₁, a₂, a₃, a₄).

$$ R^4 = \{ (a_1,a_2, a_3, a_4) \} \ \ \ \ \ \ a_1,a_2, a_3, a_4 \in R $$

In the vector space R⁴, the two binary operations (vector addition and scalar multiplication of a vector) are defined, and the operations adhere to all the properties of vector spaces.

Note. Imagining a four-dimensional or higher vector space can be challenging because we are accustomed to living in a three-dimensional geometric space. It's easier for us to conceptualize spaces with fewer dimensions (1 or 2) rather than a four-dimensional space. Nevertheless, vector spaces exist even in dimensions higher than three.

Example 5: The Vector Space Rⁿ

The vector space Rⁿ is the Cartesian product RxRx...xR, where R is multiplied by itself n times.

$$ R^n = \{ (a_1, a_2, a_3, ... , a_n) \} \ \ \ \ \ \ a_i \in R $$

Each element of the vector space Rⁿ is a vector with n components, namely an ordered n-tuple of real numbers a₁, ..., a_n.

$$ \vec{v} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \\ \vdots \\ a_n \end{pmatrix} \ \ \ \ \ \ \ \vec{v} \in V = R^n $$

Therefore, the vector space Rⁿ is the space of ordered n-tuples from the field K=R.

In the vector space Rⁿ, two binary operations are defined:

1. The addition of two vectors v₁ and v₂ results in another vector belonging to the same vector space.
   
2. The scalar multiplication of a scalar α with a vector v results in another vector belonging to the same vector space.
   

The two operations satisfy all the properties of a vector space.

Note. The properties (axioms) of a vector space are as follows: associative, commutative, distributive of multiplication over vector addition, distributive of multiplication over scalar addition, existence of a neutral element, existence of an inverse element, compatibility of scalar multiplication, compatibility of the neutral element.

Norm and Scalar Product

Vector spaces do not inherently include the notions of a vector's length and angle.

To introduce these concepts into vector spaces, an additional structure is required, which comprises two mathematical entities:

• Norm. It's the mathematical concept that defines the length (or magnitude) of a vector.
• Dot Product. It's the mathematical concept that defines the angle between two vectors.

Note. There are various structures to choose from, each with different definitions of norm and scalar product. Therefore, there isn't a single possible definition of a vector's angle and length.

Linear Combination

One of the most common operations in a vector space is the linear combination.

A linear combination is an operation involving vectors and scalars within a vector space V defined over the field K=R. $$ \vec{w} = a_1 \cdot \vec{v_1} + a_2 \cdot \vec{v_2} + a_3 \cdot \vec{v_3} + ... + a_n \cdot \vec{v_n} $$ It results in a vector (w) obtained through the sum of products between n scalar coefficients (a) and n vectors (v).

Where w, v₁, v₂, ... ,v_n are vectors in the vector space V

$$ \vec{w}, \vec{v_1}, \vec{v_2}, ... , \vec{v_n} \in V $$

While the coefficients a₁, a₂, ... , a_n are numbers (scalars), that is elements of the field of real numbers K=R.

$$ a_1, a_2, ... , a_n \in K=R $$

Linearly Dependent and Independent Vectors

Linear combinations allow us to determine which vectors in a vector space are

• Linearly Independent Vectors
  The vectors v₁, v₂,..., v_n are linearly independent if the only solution to their linear combination yielding the zero vector $$ a_1 \cdot \vec{v_1} + a_2 \cdot \vec{v_2} + ... + a_n \cdot \vec{v_n} = \vec{0} $$ is the trivial combination where all scalar coefficients are zero a_i = 0 $$ a_1 = a_2 = ... = a_n = 0 $$
• Linearly Dependent Vectors
  The vectors v₁, v₂,..., v_n are linearly dependent if there is one or more non-trivial solutions to their linear combination that results in the zero vector, where at least one scalar coefficient a_i ≠ 0 is non-zero. $$ a_1 \cdot \vec{v_1} + a_2 \cdot \vec{v_2} + ... + a_n \cdot \vec{v_n} = \vec{0} $$ In this case, not all coefficients a_i to achieve the zero vector are null. $$ \exists \ a_i \ne 0 $$

In essence, two or more vectors are linearly dependent on each other when they are not linearly independent.

Types of Vector Spaces

Vector spaces are divided into two categories:

• Finely Generated Vector Spaces
  These are vector spaces generated by a finite number of vectors.
• Infinitely Generated Vector Spaces
  These are vector spaces generated by an infinite number of vectors.

And so on.