The linear algebra is the branch of mathematics that studies concepts such as vectors , matrices , systems of linear equations and a more formal approach, vector spaces , and linear transformations. It is an active area that has connections to many areas within and outside of mathematics such as functional analysis , differential equations , operations research, computer graphics, engineering, etc.
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- 1 Introduction
- 2 General context
- 3 Founders and diversification
- 4 Applications
- 5 Source
Mathematics is, of course, a discipline. However, it is also a tool that is used in many fields. Linear algebra is a branch of modern mathematics that plays a central role because it deals with the study of concepts such as vectors, matrices, systems of linear equations, vector spaces, and linear transformations. In linear algebra, concepts are as important as calculations, so it becomes a suitable course for introducing abstract thinking, because a large part of its field has a geometric interpretation, which can help precisely visualize those concepts. .
More formally, linear algebra studies sets called vector spaces , which consist of a set of vectors and a set of scalars that have a field structure, with a vector addition operation and a product operation between scalars and vectors that satisfy certain properties for example, that the sum is commutative. It also studies linear transformations, which are functions between vector spaces that satisfy the linearity conditions: Finally, linear algebra also studies the properties that appear when additional structure is imposed on vector spaces, one of the most frequent being the existence of a product internal a kind of product between two vectors that allows us to introduce notions such as vector length and angle between a pair of them.
Founders and diversification
The history of modern linear algebra dates back to the 1843s when William Rowan Hamilton, from whom the use of the term vector comes, created the quaternions; and from 1844 when Hermann Grassmann published his book Die lineare Ausdehnungslehre (The Linear Theory of Extension). Since linear algebra is a successful theory, its methods have been developed by other areas of mathematics: in module theory , which replaces the body in the scalars with a ring; in multilinear algebra, one deals with ‘multiple variables’ in a linear mapping problem, in which each number of the different variables addresses the concept of tensor; in spectrum theory of infinite-dimensional matrix control operators, applying mathematical analysis to a theory that is not purely algebraic. In all these cases the technical difficulties are much greater.
It is an inescapable tool for the study of many other subjects that make up the curriculum design of an engineering student. It has a large number of applications in other areas, among which we can mention the space industry , electrical circuits , communication networks , archeology , weather forecasting, population movements, relativity, traffic analysis and trade routes, etc.