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## Linear algebra
See also Glossary of linear algebra.
## HistoryThe history of modern linear algebra dates back to the years 1843 and 1844. In 1843, William Rowan Hamilton (from whom the termvector stems) discovered the quaternions. In 1844, Hermann Grassmann published his book Die lineare Ausdehnungslehre ... (for the complete title see References). to be continued
k;k;## Elementary IntroductionLinear 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.
Linear algebra today has been extended to consider 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.
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## Generalisation and related topicsSince 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. ## References | |||||||

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