Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis. Linear algebra also has a concrete representation in analytic geometry. It has extensive applications in the natural sciences and the social sciences, since nonlinear models can often be approximated by a linear model.
See also list of linear algebra topics.
The history of modern linear algebra dates back to the years 1843 and 1844. In 1843, William Rowan Hamilton (from whom the term vector stems) discovered the quaternions. In 1844, Hermann Grassmann published his book Die lineare Ausdehnungslehre (see References).
Linear algebra had its beginnings in the study of vectors in Cartesian 2-space and 3-space. A vector, here, is a directed line segment, characterized by both length or magnitude and direction. Vectors can be used then to represent certain physical entities such as forces, and they can be added and multiplied with scalars, thus forming the first example of a real vector space.
Modern Linear algebra has been extended to consider spaces of arbitrary or infinite dimension. A vector space of dimension n is called an n-space. Most of the useful results from 2 and 3-space can be extended to these higher dimensional spaces. Although many people cannot easily visualize vectors in n-space, such vectors or n-tuples are useful in representing data. Since vectors, as n-tuples, are ordered lists of n components, most people can summarize and manipulate data efficiently in this framework. For example, in economics, one can create and use, say, 8-dimensional vectors or 8-tuples to represent the Gross National Product of 8 countries. One can decide to display the GNP of 8 countries for a particular year, where the countries' order is specified, for example, (United States, United Kingdom, France, Germany, Spain, India, Japan, Australia), by using a vector (v1, v2, v3, v4, v5, v6, v7, v8) where each country's GNP is in its respective position.
A vector space (or linear space), as a purely abstract concept about which we prove theorems, is part of abstract algebra, and well integrated into this field. Some striking examples of this are the group of invertible linear maps or matrices, and the ring of linear maps of a vector space. Linear algebra also plays an important part in analysis, notably, in the description of higher order derivatives in vector analysis and the study of tensor products and alternating maps.
A vector space is defined over a field, such as the field of real numbers or the field of complex numbers. Linear operators take elements from a linear space to another (or to itself), in a manner that is compatible with the addition and scalar multiplication given on the vector space(s). The set of all such transformations is itself a vector space. If a basis for a vector space is fixed, every linear transform can be represented by a table of numbers called a matrix. The detailed study of the properties of and algorithms acting on matrices, including determinants and eigenvectors, is considered to be part of linear algebra.
One can say quite simply that the linear problems of mathematics - those that exhibit linearity in their behaviour - are those most likely to be solved. For example differential calculus does a great deal with linear approximation to functions. The difference from non-linear problems is very important in practice.
The general method of finding a linear way to look at a problem, expressing this in terms of linear algebra, and solving it, if need be by matrix calculations, is one of the most generally applicable in mathematics.
Some useful theorems
Equivalent statements for square matrices
A theorem in linear algebra states that for any n by n matrix A, all the following statements are equivalent (either always true or always false):
- A is invertible.
- A does not have 0 as an eigenvalue.
- For any b Fn, Ax=b exactly one solution for x.
- Ax=0 only has the trivial solution.
- ATA is invertible.
- A is row (column) equivalent to the identity matrix.
- The rows and columns of A span Fn.
- The null space of A contains only the zero vector.
- The range of A is Fn.
- The rows (columns) of A form a linearly independent set of vectors in Fn (Fn).
Where F is the field the matrix entries are from. Often, this field is the real numbers or the complex numbers.
Generalization and related topics
Since linear algebra is a successful theory, its methods have been developed in other parts of mathematics. In module theory one replaces the field of scalars by a ring. In multilinear algebra one deals with the 'several variables' problem of mappings linear in each of a number of different variables, inevitably leading to the tensor concept. In the spectral theory of operators control of infinite-dimensional matrices is gained, by applying mathematical analysis in a theory that isn't purely algebraic. In all these cases the technical difficulties are much greater.
- Grassmann, Hermann, Die lineare Ausdehnungslehre dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie, 1844.