linear algebra

It is called algebra to the branch of math that is oriented to the generalization of arithmetic operations via signs, letters and numbers. In algebra, letters and signs represent another entity through symbolism.

Linear, for its part, 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 that which has consequences that are proportional to a cause.

Linear algebra is known as algebra specialization who works with arrays, vector, Vector spaces Y linear 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 are structures that arise when a set that is not empty, an external operation and an internal operation is registered. The vector 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 have the systems of linear equations.

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 numbersAs with 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 the quaternion must satisfy certain rules of addition, multiplication and equality. This discovery was of considerable importance for mathematics. With respect to set of the real numbers, it is defined as the one in which the rational numbers (zero, positive and negative) and irrational numbers (those that cannot be expressed) are found.

Continuing with the definition of the elements that linear algebra deals with, it is important to know that a system of linear equations It 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 that are used to carry out the description of a phenomenon with magnitude, though 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 starting from any set over a fixed field.

Another of the points of interest of linear algebra is the group of properties that appears when it is imposed on top of vector spaces structure additional; a very frequent example of this occurs when a Internal product, that is, 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, the Functional analysis, the engineering, the operations research and the computer graphics. Likewise, areas of mathematics such as module theory or the multilinear algebra they have been developed from linear algebra.