mathematical discipline that deals with vectors (vector) and matrices (matrix) and, more generally, with vector spaces (vector space) and linear transformations. Unlike other parts of mathematics that are frequently invigorated by new ideas and unsolved problems, linear algebra is very well understood. Its value lies in its many applications, from mathematical physics to modern algebra (algebra, modern) and coding theory.

Representing vectors as arrows in two or three dimensions is a starting point, but linear algebra has been applied in contexts where this is no longer appropriate. For example, in some types of differential equations (differential equation) the sum of two solutions gives a third solution, and any constant multiple of a solution is also a solution. In such cases the solutions can be treated as vectors, and the set of solutions is a vector space in the following sense. In a vector space any two vectors can be added together to give another vector, and vectors can be multiplied by numbers to give “shorter” or “longer” vectors. The numbers are called scalars (scalar) because in early examples they were ordinary numbers that altered the scale, or length, of a vector. For example, if v is a vector and 2 is a scalar, then 2v is a vector in the same direction as v but twice as long. In many modern applications of linear algebra, scalars are no longer ordinary real numbers (real number), but the important thing is that they can be combined among themselves by addition, subtraction, multiplication, and division. For example, the scalars may be complex numbers (complex number), or they may be elements of a finite field such as the field having only the two elements 0 and 1, where 1+1=0. The coordinates of a vector are scalars, and when these scalars are from the field of two elements, each coordinate is 0 or 1, so each vector can be viewed as a particular sequence of 0s and 1s. This is very useful in digital processing, where such sequences are used to encode and transmit data.

Vector spaces are one of the two main ingredients of linear algebra, the other being linear transformations (or “operators” in the parlance of physicists). Linear transformations are functions (function) that send, or “map,” one vector to another vector. The simplest example of a linear transformation sends each vector to c times itself, where c is some constant. Thus, every vector remains in the same direction, but all lengths are multiplied by c. Another example is a rotation, which leaves all lengths the same but alters the directions of the vectors. Linear refers to the fact that the transformation preserves vector addition and scalar multiplication. This means that if T is a linear transformation sending a vector v to T(v), then for any vectors v and w, and any scalar c, the transformation must satisfy the properties T(v+w)=T(v)+T(w) and T(cv)=cT(v).

When studying linear transformations, it is extremely useful to find nonzero vectors whose direction is left unchanged by the transformation. These are called eigenvectors (also known as characteristic vectors). If v is an eigenvector for the linear transformation T, then T(v)=λv for some scalar λ. This scalar is called an eigenvalue. The eigenvalue of greatest absolute value, along with its associated eigenvector, have special significance for many physical applications. This is because whatever process is represented by the linear transformation often acts repeatedly—feeding output from the last transformation back into another transformation—which results in every arbitrary (nonzero) vector converging on the eigenvector associated with the largest eigenvalue, though rescaled by a power of the eigenvalue. In other words, the long-term behaviour of the system is determined by its eigenvectors.

Finding the eigenvectors and eigenvalues for a linear transformation is often done using matrix algebra, first developed in the mid-19th century by the English mathematician Arthur Cayley (Cayley, Arthur). His work formed the foundation for modern linear algebra.