Vector space

Vector add scale

Vector components and base change

Example for addition of functions

Vector components

Matrix

Determinant parallelepiped

Vector space is a fundamental concept in mathematics, particularly in the fields of linear algebra and abstract algebra. It provides a framework for studying various algebraic structures and is widely applicable in areas such as physics, engineering, and computer science.

Definition

A vector space over a field F consists of a set V along with two operations: vector addition and scalar multiplication. The elements of V are called vectors, and the elements of F are called scalars. For V to be a vector space, the following conditions must be satisfied for all vectors u, v, and w in V, and all scalars a and b in F:

1. Vector addition is commutative: u + v = v + u. 2. Vector addition is associative: (u + v) + w = u + (v + w). 3. There exists an element 0 in V, called the zero vector, such that u + 0 = u for all u in V. 4. For each u in V, there exists an element -u in V, called the additive inverse of u, such that u + (-u) = 0. 5. Scalar multiplication is distributive with respect to vector addition: a(u + v) = a'u + a'v. 6. Scalar multiplication is distributive with respect to field addition: (a + b)u = a'u + b'u. 7. Scalar multiplication is associative with respect to field multiplication: a(b'u) = (a'b)u. 8. Multiplication by 1 (the multiplicative identity in F) leaves vectors unchanged: 1u = u.

Examples

1. The set of all two-dimensional vectors (x, y), where x and y are real numbers, forms a vector space over the field of real numbers with the usual operations of vector addition and scalar multiplication. 2. The set of all n-tuples of real numbers is a vector space over the real numbers. 3. The set of all polynomials with coefficients in F forms a vector space over F.

Basis and Dimension

A basis of a vector space V is a set of vectors in V that is linearly independent and spans V. The dimension of V is the number of vectors in a basis of V, which is a well-defined quantity.

Subspaces

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

Linear Transformations

A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.

Applications

Vector spaces are used in various scientific fields. In physics, they are used to describe physical quantities such as forces and velocities. In computer science, they are used in algorithms and data structures, particularly those involving graphics and machine learning.