It is called algebra to the branch of mathematics which is oriented to 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.
It is known as linear algebra to the algebra specialization who works with matrices , vectors , Vector spaces and linear type equations . It is an area of knowledge that developed especially in the 1840s with the contributions of German Hermann Grassmann (1809-1877) and the Irish William Rowan Hamilton (1805-1865), among other mathematicians.
The Vector spaces they are structures that arise when a set that is not empty, an external operation and an internal operation are registered. The vectors they 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.
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 with the complexes, and these are 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 quaternion must meet certain rules of addition, multiplication and equality . This discovery was of considerable importance to mathematics. With respect to 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 it is composed, as the 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 another of scalars. The scalar they are elements of the mathematical bodies that are used to carry out the description of a phenomenon with magnitude, but without direction; It can be a real, complex or constant number.
In linear transformations, vectors are not always successions of scalars; 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 appear when the vector spaces are imposed structure additional; a very frequent example of this takes place when a Internal product , that is to say, a kind of product between a pair of vectors, which gives rise to 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 , he Functional analysis , the engineering , the operations research and the computer graphics . Also, areas of mathematics such as the module theory or the multilineal algebra They have been developed from linear algebra.