Mathematics\Linear Algebra\Vector Spaces

Description

Vector spaces form the foundational concept upon which much of linear algebra is built. A vector space (also referred to as a linear space) is a collection of objects called vectors, which can be added together and multiplied (scaled) by numbers, called scalars. Scalars are commonly real numbers, but they can also come from other fields such as complex numbers.

Definition and Axioms

Formally, a vector space \( V \) over a field \( F \) is defined by the following properties and operations:

Addition: For any vectors \( \mathbf{u}, \mathbf{v} \in V \), there is a vector \( \mathbf{w} \in V \) denoted as \( \mathbf{u} + \mathbf{v} \).
Scalar Multiplication: For any scalar \( c \in F \) and vector \( \mathbf{u} \in V \), there is a vector \( c\mathbf{u} \in V \).

These operations must satisfy the following axioms for all \( \mathbf{u}, \mathbf{v}, \mathbf{w} \in V \) and all scalars \( a, b \in F \):

Commutativity of Addition: \( \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \).
Associativity of Addition: \( (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}) \).
Additive Identity: There exists an element \( \mathbf{0} \in V \) such that \( \mathbf{u} + \mathbf{0} = \mathbf{u} \) for all \( \mathbf{u} \in V \).
Additive Inverse: For each \( \mathbf{u} \in V \), there exists an element \( -\mathbf{u} \in V \) such that \( \mathbf{u} + (-\mathbf{u}) = \mathbf{0} \).
Distributivity of Scalar Multiplication (over vector addition): \( a(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v} \).
Distributivity of Scalar Multiplication (over field addition): \( (a + b)\mathbf{u} = a\mathbf{u} + b\mathbf{u} \).
Associativity of Scalar Multiplication: \( a(b\mathbf{u}) = (ab)\mathbf{u} \).
Multiplicative Identity of Scalars: \( 1\mathbf{u} = \mathbf{u} \), where 1 denotes the multiplicative identity in the field \( F \).

Examples

Euclidean Space: The set \( \mathbb{R}^n \), consisting of all \( n \)-tuples of real numbers, is a vector space over the field \( \mathbb{R} \) of real numbers.
\[
\text{For instance, in } \mathbb{R}^2, \text{ let } \mathbf{u} = \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} \text{ and } \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}. \text{ Then } \mathbf{u} + \mathbf{v} = \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \end{pmatrix}.
\]
Polynomial Space: The set of polynomials with coefficients in a field \( F \), denoted \( F[x] \), is a vector space over \( F \). The vectors are the polynomials, and addition and scalar multiplication are defined as usual for polynomials.

Basis and Dimension

A set of vectors \( \{ \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \} \) in a vector space \( V \) is called a basis if:

The set is linearly independent, meaning no vector in the set can be expressed as a linear combination of the others.
The set spans the vector space \( V \), meaning any vector in \( V \) can be expressed as a linear combination of the basis vectors.

The number of vectors in the basis is called the dimension of the vector space, denoted as \( \dim(V) \).

Linear Transformations

A linear transformation is a function \( T: V \rightarrow W \) between two vector spaces \( V \) and \( W \) that preserves the operations of vector addition and scalar multiplication:

\( T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}) \).
\( T(a\mathbf{u}) = aT(\mathbf{u}) \).

Linear transformations can often be represented by matrices, which makes the study of vector spaces closely related to matrix theory and systems of linear equations.

Applications

Vector spaces are ubiquitous in both pure and applied mathematics. They play critical roles in:

Solving systems of linear equations.
Transforming geometric objects.
Representing and manipulating data in computer science.
Modeling physical phenomena in engineering and physics.
Encoding and transmitting data in telecommunications through vector spaces over finite fields.

In summary, vector spaces provide a fundamental framework for understanding and working with a wide range of mathematical and real-world problems.