Vector space

In mathematics and science, a vector space is a set of elements, called vectors, for which the operations of vector addition and scalar multiplication are defined. The two processes must meet certain conditions, known as axioms.

Vector spaces are an abstract concept in linear algebra. A vector space is made up of four components: a set (V) of vectors; a field (F), which is a set of numbers called scalars; vector addition (represented by +); and scalar multiplication (represented by •). In vector addition, vectors in the vector space are added together. In scalar multiplication, vectors in the vector space are multiplied by scalars.

Vector spaces play an important role in studying linear equations. The most basic example of a vector space is the Euclidian space. Vector spaces are used in problem-solving applications in the subjects of science, engineering, and mathematics. They are used in real-life applications in the fields of engineering, computer technology, business, and aeronautics.

Background

The concept of vector spaces is a complex one, and it took time before the notion gained acceptance in the study of mathematics. The idea was introduced by German mathematician Hermann Grassmann in 1844. However, the obscure terminology Grassmann used to describe his theories made his work hard to understand. Grassmann later clarified his ideas in 1862.

Italian mathematician Giuseppe Peano refined Grassmann's work and established the axioms used to define vector space in 1888. Peano's axiom system gained prominence when German mathematician Hermann Weyl referenced them in 1918.

Vectors are quantities that have the dimensions of size and direction. A vector can be drawn as a directional line segment with an arrowhead. The length of the line depicts its size, or magnitude, while the arrowhead shows its direction. Examples of vectors include velocity, force, acceleration, and momentum.

A scalar is a quantity that has only one dimension: size. It is just a number, or value. Scalars can be real numbers or complex numbers. When they are multiplied against a vector, they "scale" the vector up or down in size. Examples of scalars include length, area, mass, and speed.

The elements that comprise the vector space V are not specified, though for general purposes they are called vectors.

The vector space and field of scalars are written together as the "vector space V over the field F." When the field of scalers is made up of real numbers, the vector space is known as a real vector space and referred to as the "vector space over real numbers." When the field of scalars is comprised of complex numbers, the vector space is known as a complex vector space and referred to as the "vector space over complex numbers."

The most basic vector space is the Euclidian space, a two- or three-dimensional space whose elements are made up of the list of all real numbers. All the scalars are also real numbers. Addition is carried out component-wise, and each component is multiplied by the scalar.

Vector spaces can be used to solve linear equations when two expressions are set equal to one other and to solve the addition and multiplication of matrices, which are rectangular displays of numbers.

Overview

Ten generally accepted axioms determine a vector space. Five properties pertain to vector addition, and five properties concern scalar multiplication. In the examples that follow, x, y, and z are any vectors in the vector space V. The vector space V is over the field F, with F representing the scalar.

Vector addition must satisfy the following axioms.

• Closure: If x and y are vectors in V, then their sum, x + y, is also contained in V.
• Commutativity: When x and y are added together, their sum is the same no matter what order the vectors are in (e.g., x + y = y + x).
• Associativity: When x, y, and z, are added together, it does not matter how the vectors are grouped as the sum will stay the same (e.g., x + (y + z) = (x + y) + z).
• Additive identity: In V, there exists a zero vector, 0, such that when it is added to any vector, the sum will be the vector itself (e.g., 0 + x = x + 0 = x).
• Additive inverse: For any vector x in V, there exists an inverse of that vector, –x. When the vector and its inverse are added together, the sum will be zero (e.g., x + (–x) = 0).

Scalar multiplication must adhere to the following axioms.

• Closure: If x is a vector in V, and F is the scalar, then the product x • F is contained within V.
• Distributivity (vector addition): The sum of x and y multiplied by F is equal to the sum of each vector multiplied by F (e.g., F • (x + y) = Fx + Fy).
• Distributivity (scalar addition): The sum of F and another scalar, G, multiplied by x is equal to the sum of each scalar multiplied by x (e.g., (F + G) • x = Fx + Gx).
• Associativity: When F, G, and x are multiplied together, it does not matter how they are grouped as the product will remain the same (e.g., F • (Gx) = (FG) • x).
• Unitary: When a vector is multiplied by 1, the product will be the vector itself (e.g., 1 • x = x).

When one vector space lies inside another one, it is a subset, W, of vector space V. If the subset meets all axioms for vector addition and scalar multiplication, then the subset is a subspace of the vector space V.

The subset W must also obey three additional requirements to be considered a subspace: W must be nonempty, meaning it contains the zero vector and does not equal zero; it is closed under vector addition; and it is closed under scalar multiplication.

Vector spaces are studied in algebra, calculus, physics, mechanical engineering, electrical engineering, business, and computer science. In real-world applications, vector spaces are used throughout the professions of engineering, computers, business, aeronautics, and various fields of science.

Bibliography

Corwin, Lawrence J., and Robert H. Szczarba. Calculus in Vector Spaces. 2nd ed., Marcel Dekker Inc., 1995.

Halmos, Paul R. Finite Dimensional Vector Spaces. Princeton UP, 1948.

"Math 240: Vector Spaces." Northeastern Illinois University, www.math.niu.edu/~beachy/courses/240/06spring/vectorspace.html. Accessed 30 Jan. 2017.

"Multiplication Properties." AAA Math, www.aaamath.com/pro74b-propertiesmult.html. Accessed 3 Jan. 2023.

Poole, David. Linear Algebra: A Modern Introduction. Cengage Learning, 2014.

"Scalars and Vectors." NASA, www.grc.nasa.gov/WWW/K-12/airplane/vectors.html. Accessed 30 Jan. 2017.

Scott, Geoffrey. "What Is a Vector Space?" University of Toronto, www.math.toronto.edu/gscott/WhatVS.pdf. Accessed 30 Jan. 2017.

Stillwell, John. Elements of Mathematics: From Euclid to Godel. Princeton UP, 2016.

"Vector Space." Wolfram MathWorld, 2023, mathworld.wolfram.com/VectorSpace.html. Accessed 3 Jan. 2023

"Vector Spaces." University of Miami, www.physics.miami.edu/~nearing/mathmethods/vector‗spaces.pdf. Accessed 30 Jan. 2017.

"Vector Spaces." University of Puget Sound, linear.ups.edu/html/section-VS.html. Accessed 30 Jan. 2017.

Ward Cheney, Elliott, and David Ronald Kincaid. Linear Algebra: Theory and Applications. Jones and Bartlett Publishers, 2009.