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See also list of linear algebra topics.
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).
Linear algebra had its beginnings in the study of vectors in CartesianCartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. This work was influential to the development of analytic geometry, calculus, and cartography. The idea 2-space and 3-space. A vector, here, is a directed line segmentIn mathematics, a line segment is a part of a line that is bounded by two end points. See also interval (mathematics). When the end points both lie on a circle, a line segment is called a chord. When they are both vertices of a polygon, it is an edge if t, characterized by both length or magnitude and direction. Vectors can be used then to represent certain physical entities such as forceIn physics, a net force acting on a body causes that body to accelerate; that is, to change its velocity. The concept appeared first in the second law of motion of classical mechanics. It is usually expressed by the equation F m · a where F is the force,s, and they can be added and multiplied with scalarAbstract algebra Algebra Linear algebra The concept of a scalar is used in mathematics and physics. The concept used in physics is a more concrete version of the same idea that goes by that name in mathematics. In mathematics, the meaning of scalar depends, thus forming the first example of a realIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers may 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-tuplesSee also tuple (music) as in duple and triple. In mathematics, a tuple is a finite sequence of objects (an ordered list of a limited number of objects). An infinite sequence is a family. Tuples are used by mathematicians to describe mathematical objects t 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 economicsEconomics is the social science studying how society uses its limited resources to meet desires and wants. Put otherwise, economics studies what, how and for whom society produces. This involves analyzing the production, distribution and consumption of go, 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 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.