Linear algebra

Linear subspaces with shading

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear mappings that operate on these spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces. Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of linear algebra, while it is also a necessary tool in the study of analytical geometry, engineering, physics, computer science, the social sciences, and the natural sciences.

Overview

Linear algebra's primary objects of study are vectors, which are directed quantities that can be added together and multiplied ("scaled") by numbers, called scalars in this context. Scalars are often taken from the real numbers, but one may also consider vectors with complex number components. The operations of vector addition and scalar multiplication must satisfy certain requirements, known as axioms, listed below under Vector Spaces.

Vector Spaces

A vector space (also called a linear space) involves a collection of vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are elements of a predefined field, which is most commonly the field of real numbers or complex numbers. The basic axioms that vector spaces must satisfy are:

• Associativity of addition
• Commutativity of addition
• Identity element of addition
• Inverse elements of addition
• Compatibility of scalar multiplication with field multiplication
• Identity element of scalar multiplication
• Distributivity of scalar multiplication with respect to vector addition
• Distributivity of scalar multiplication with respect to field addition

Matrices

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries. Matrices are a key tool in linear algebra, used to represent and solve systems of linear equations, perform linear transformations, and more. Operations such as matrix addition, matrix multiplication, and finding the inverse of a matrix are fundamental in many areas of mathematics and its applications.

Determinants and Eigenvalues

The determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the matrix. It is used in analytical geometry, for solving linear equations, and for determining the invertibility of matrices.

Eigenvalues and eigenvectors are concepts related to linear transformations represented by matrices. An eigenvector of a matrix is a non-zero vector that only changes by a scalar factor when that matrix is applied to it. The corresponding scalar factor is called an eigenvalue. Finding the eigenvalues and eigenvectors of a matrix is fundamental in various areas of mathematics and its applications, including systems theory and quantum mechanics.

Applications

Linear algebra is not just a subject of abstract interest. It is used extensively in applied mathematics, engineering, physics, computer science, economics, statistics, and many other disciplines. The development of computer algorithms for linear algebra computations is a subject of ongoing research, with new methods for solving systems of linear equations and matrix factorizations being developed.

See Also

• Matrix theory
• Vector space
• Eigenvalues and eigenvectors
• System of linear equations
• Determinant