It is called algebra to the branch of mathematics that targets the generalization of arithmetic operations through signs, letters and numbers. In algebra, letters and signs represent another entity through symbolism.
Linear, according to Digopaul, 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 the specialization of algebra that works with matrices, vectors, vector spaces, and linear equations. It is an area of knowledge that was developed especially in the 1840s with contributions from the German Hermann Grassmann (1809-1877) and the Irish William Rowan Hamilton (1805–1865), among other mathematicians.
The vector spaces are structures that arise when a set is not empty, an external operation and an operation internal registers. The vectors are the elements that are part of the vector space. Regarding matrices, it is a two-dimensional set of numbers that allow the representation of the coefficients that linear systems of equations have.
William Rowan Hamilton is one of the most prominent names in the field of mathematics, since he was the one who coined the term “vector”, in addition to having created the quaternions. This concept extends from real numbers, as it does with complexes, and groups of four numbers are very useful when studying quantities in three dimensions that expect to have a magnitude and a direction.
The numbers that make up the 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 that in which the rational ones are (zero, positive and negative) and the irrational ones (those that cannot be expressed).
Continuing with the DEFINITION OF the elements with which linear algebra deals, it is important to know that a system of linear equations is composed, as its name implies, of linear equations (a set of equations that are of the 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 the other of scalars. The scalars are elements of mathematical bodies used to carry out the description of a phenomenon magnitude but not 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 point of interest in linear algebra is the group of properties that appears when additional structure is imposed on 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 leads to the introduction of concepts such as the angle between 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.. Likewise, areas of mathematics such as module theory or multilinear algebra have been developed from linear algebra.