Linear algebra

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Linear Algebra is the study of vectors and vector spaces, structure preserving maps between vector spaces, and extensions of these concepts. Vector spaces are a central unifying theme of modern mathematics, and so linear algebra is firmly rooted in both abstract algebra and analysis. Furthermore, it has a concrete representation in analytic geometry and it has extensive applications in the pure sciences and the social sciences.

A vector 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.

Linear algebra had its beginnings in the study of vectors in real 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 real numbers, thus forming the first example of a real vector space.

Linear algebra today has been extended to consider n-space, since most of the useful results from 2 and 3-space can be extended to n-dimensional space. Although one cannot 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, one 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.

But, to begin at the beginning, one has to define some "elementary" objects and properties on which linear algebra is built and look at some examples. Included here are:

A vector space (or linear 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.