linear algebra

A line passing through the origin (blue, thick) in R³ is a linear subspace, a common object of study in linear algebra.

Linear algebra is a branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called 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 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 linear ones.

History

Many of the basic tools of linear algebra, particularly those concerned with the solution of systems of linear equations, date to antiquity--- see e.g. the history of Gaussian elimination--- although many objects were not isolated and considered in their own right until the 1600s and 1700s (see the history of determinants). The method of least squares, first used by Gauss in the 1790s, is an early and significant application of the ideas of linear algebra.

The subject began to take its modern form in the mid-19th century, which saw many notions and methods of previous centuries abstracted and generalized as the beginnings of abstract algebra. Matrices and tensors were introduced as abstract mathematical objects and well studied by the turn of the 20th century. The use of these objects in special relativity, statistics, and quantum mechanics did much to spread the subject beyond pure mathematics.

Remarkably, the 2 × 2 complex matrices were studied before 2 × 2 real matrices. The early interest was expressed in terms of biquaternions and Pauli algebra. Investigation of the 2 × 2 real matrices revealed the less common split-complex numbers and dual numbers which are at variance with the Euclidean nature of the ordinary complex number plane.

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 (also called length or norm) and its direction. The zero vector is an exception; it has zero magnitude and no direction. Vectors can be used to represent physical entities such as forces, and they can be added to each other and multiplied by scalars, thus forming the first example of a real vector space, where a distinction is made between "scalars", in this case real numbers, and "vectors".

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 people cannot easily visualize vectors in n-space, such vectors or n-tuples are useful in representing data. Since vectors, as n-tuples, consist of n ordered components, data can be efficiently summarized and manipulated 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, Italy, Japan, Switzerland and Belgium), 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 theorems are proved, is part of abstract algebra, and is well integrated into this discipline. 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.

In this abstract setting, the scalars with which an element of a vector space can be multiplied need not be numbers. The only requirement is that the scalars form a mathematical structure, called a field. In applications, this field is usually the field of real numbers or the field of complex numbers. Linear maps 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.

Some useful theorems

• Every vector space has a basis.^[1]
• Any two bases of the same vector space have the same cardinality; equivalently, the dimension of a vector space is well-defined.^[2]
• A matrix is invertible if and only if its determinant is nonzero.^[3]
• A matrix is invertible if and only if the linear map represented by the matrix is an isomorphism.
• If a square matrix has a left inverse or a right inverse then it is invertible (see 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.
• An n×n matrix is diagonalizable (i.e. there exists an invertible matrix P and a diagonal matrix D such that A = PDP⁻¹) if and only if it has n linearly independent eigenvectors.
• The spectral theorem states that a matrix is orthogonally diagonalizable if and only if it is symmetric.

For more information regarding the invertibility of a matrix, consult the invertible matrix article.

Generalizations 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 considers multivariable linear transformations, that is, mappings which are linear in each of a number of different variables. This line of inquiry naturally leads to the idea of the tensor product. Functional analysis mixes the methods of linear algebra with those of mathematical analysis.

See also

• List of linear algebra topics
• Numerical linear algebra

Notes

1. ^ The existence of a basis is straightforward for finitely generated vector spaces, but in full generality it is logically equivalent to the axiom of choice.
2. ^ Dimension theorem for vector spaces
3. ^ Pragma's Playground: Matrices for Dummies, http://www.pragmaware.net/articles/matrices/index.php

References

This article includes a list of references or external links, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations where appropriate. (September 2009)

Textbooks

• Axler, Sheldon Jay (1997), Linear Algebra Done Right (2nd ed.), Springer-Verlag, ISBN 0387982590 
• Lay, David C. (August 22, 2005), Linear Algebra and Its Applications (3rd ed.), Addison Wesley, ISBN 978-0321287137 
• Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), ISBN 978-0898714548, http://www.matrixanalysis.com/DownloadChapters.html 
• Poole, David (2006), Linear Algebra: A Modern Introduction (2nd ed.), Brooks/Cole, ISBN 0-534-99845-3 
• Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International 
• Leon, Steven J. (2006), Linear Algebra With Applications (7th ed.), Pearson Prentice Hall 

History

• Fearnley-Sander, Desmond, "Hermann Grassmann and the Creation of Linear Algebra" (via JSTOR), 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.

Further reading

External links

Wikibooks has more on the topic of Linear algebra

• International Linear Algebra Society
• MIT Professor Gilbert Strang's Linear Algebra Course Homepage : MIT Course Website
• MIT Linear Algebra Lectures: free videos from MIT OpenCourseWare
• Linear Algebra Toolkit.
• Linear Algebra on MathWorld.
• Linear Algebra overview and notation summary on PlanetMath.
• Matrix and Linear Algebra Terms on Earliest Known Uses of Some of the Words of Mathematics
• Earliest Uses of Symbols for Matrices and Vectors on Earliest Uses of Various Mathematical Symbols
• 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. José Figueroa-O'Farrill, University of Edinburgh
• Online Notes / Linear Algebra Paul Dawkins, Lamar University
• Elementary Linear Algebra textbook with solutions
• Linear Algebra Wiki
• Linear algebra (math 21b) homework and exercises

Online books

• Beezer, Rob, A First Course in Linear Algebra
• Connell, Edwin H., Elements of Abstract and Linear Algebra
• Hefferon, Jim, Linear Algebra
• Matthews, Keith, Elementary Linear Algebra