Linear algebra
See also list of linear algebra topics.
Table of contents |
2 Elementary introduction 3 Some useful theorems 4 Generalisation and related topics 5 See also 6 external link 7 References |
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 lineare Ausdehnungslehre (see References).
Elementary introduction
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 (v_{1}, v_{2}, v_{3}, v_{4}, v_{5}, v_{6}, v_{7}, v_{8}) 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
- Every linear space has a basis.
- A matrix A with n rows and n columns is non-singular if there exists a matrix B that satisfies AB = BA = I where I is the identity matrix.
- A matrix is non-singular if and only if (iff) its determinant is different than zero.
- A matrix is non-singular iff the linear transformation represented by the matrix is isomorphism.
- to be added
Generalisation 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.
See also
external link
Linear Algebra ToolkitReferences
- 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.
- Hermann Grassmann and the Creation of Linear Algebra
Topics in mathematics related to structure |
Abstract algebra | Number theory | Algebraic geometry | Group theory | Monoids | Analysis | Topology | Linear algebra | Graph theory | Universal algebra | Category theory |
Topics in mathematics related to spaces |
Topology | Geometry | Trigonometry | Algebraic geometry | Differential geometry and topology | Algebraic topology | Linear algebra | Fractal geometry | Compact space |