Linear algebra

Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. 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 and it is generalized in operator theory. It has extensive applications in the natural sciences and the social sciences, since nonlinear models can often be approximated by a linear model.

History

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 lineale Ausdehnungslehre (see References). Arthur Cayley introduced matrices, one of the most fundamental linear algebraic ideas, in 1857. These early references belie the fact that linear algebra is mainly a development of the twentieth century: the number-like objects matrices were hard to place before the development of ring theory in abstract algebra. With the coming of special relativity many practioners gained appreciation of the subtleties of linear algebra. Furthermore, the routine application of Cramer's rule to solve Partial differential equations led to inclusion of linear algebra in standard coursework at universities. For instance, E.T. Copson wrote:

Elementary introduction

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 its magnitude, represented by length, and its direction. Vectors can be used to represent physical entities such as forces, and they can be added to each other 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, it is possible to 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 behavior - are those most likely to be solved. For example differential calculus does a great deal with linear approximation to functions. The difference from nonlinear 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

Every linear space has a basis. (This statement is logically equivalent to the axiom of choice.)
A matrix is invertible if and only if its determinant is nonzero.
A matrix is invertible if and only if the linear transformation represented by the matrix is an isomorphism (see also invertible matrix for other equivalent statements).
A matrix is positive semidefinite if and only if each of its eigenvalues is greater than or equal to zero.
A matrix is positive definite if and only if each of its eigenvalues is greater than zero.
The spectral theorem (regarding diagonal matrices).

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 is not purely algebraic. In all these cases the technical difficulties are much greater.

References

Beezer, Rob, A First Course in Linear Algebra, licensed under GFDL.
Fearnley-Sander, Desmond, Hermann Grassmann and the Creation of Linear Algebra, American Mathematical Monthly 86 (1979), pp. 809–817.
Grassmann, Hermann, Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik: dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert, O. Wigand, Leipzig, 1844.
Jim Hefferon: Linear Algebra (Online textbook)
Edwin H. Connell: Elements of Abstract and Linear Algebra (Online textbook)

External links

MIT Linear Algebra Lectures - Videos from MIT OpenCourseWare, free to everyone
Linear Algebra Toolkit.
Linear Algebra Workbench: multiply and invert matrices, solve systems, eigenvalues etc].
Linear Algebra on MathWorld.
Linear Algebra overview and notation summary on PlanetMath.
Linear Algebra by Elmer G. Wiens. Interactive web pages for vectors, matrices, linear equations, etc.
Linear Algebra Solved Problems: Interactive forums for discussion of linear algebra problems, from the lowest up to the hardest level (Putnam).
Linear Algebra for Informatics.