Algebra is called the branch of mathematics that focuses on the generalization of arithmetic operations through signs, letters and numbers. In algebra, letters and signs represent another entity through symbolism.
Linear, meanwhile, is an adjective that refers to what is linked to a line (a line or a sequence). In the field of mathematics, the idea of linear refers to what has consequences that are proportional to a cause.
Linear algebra is known as the specialization of algebra that works with matrices, vectors, vector spaces and linear type equations.
Vector spaces are structures that arise when a set that is not empty, an external operation and an internal operation are registered. Vectors are the elements that are part of the vector space. As for the matrices, it is a two-dimensional set of numbers that allow the representation of the coefficients that the systems of linear equations have.
This concept extends from real numbers, as well as with complexes, and these are groups of four numbers that are very useful when studying quantities in three dimensions that expect to have a magnitude and a direction.
The numbers that make up quaternion must satisfy certain rules of addition, multiplication and equality. This discovery was of considerable importance to mathematics. With respect to the set of real numbers, it is defined as the one in which the rational (zero, positive and negative) and irrational (those that cannot be expressed) are found.
Following the definition of the elements with which linear algebra deals, it is important to know that a system of linear equations is composed, as the name implies, of linear equations (a set of equations that are first degree), defined on a commutative ring or a body.
Vector spaces, the focus of study of linear algebra, have two sets: one of vectors and another of scalars. Scalars are elements of the mathematical bodies that are used to carry out the description of a phenomenon with magnitude, although without direction; It can be a real, complex or constant number.
In linear transformations, vectors are not always scalar sequences; It is also possible that they are elements of any set. So much so that a vector space can arise from any set on a fixed field.
Another of the points of interest of linear algebra is the group of properties that appears when additional structure is imposed on top of the vector spaces; A very frequent example of this occurs when an internal product is presented, that is, a kind of product between a pair of vectors, which results in the introduction of concepts such as the angle formed by two vectors or their length .
It is correct to say that linear algebra is an active area that connects with many others, some of which do not belong to mathematics, such as differential equations, functional analysis, engineering, operations research and computer graphics. . Also, areas of mathematics such as module theory or multilinear algebra have been developed from linear algebra.