Lesson 8

Linear algebra is a part of the algebra that studies the theory of linear equations, matrices, determinants, vector spaces, linear transformations.

Historically, the first question of linear algebra was to find a solution of linear equations.

Linear equations as the equation of lines and planes became the subject of study after the invention of the method of Cochran Descartes and Fermat (about 1636). Hamilton (1833) introduced the term "vector". Kelly developed the theory of matrices (1850). Systems of linear equations first appeared in the works of Lagerra (1867). Grassman studied the theory of systems of algebra in 1844 and 1862.

Vector (or linear space) is called the set of vectors, which include vectors with any possible component value.

In order for a plurality of vectors to be vector space on it must act a number of axioms : commutativity, associativity, distributivity, addition and multiplication on the scalar, the existence of the zero and the opposite element.

The number n, which determines the number of vector elements is called the dimension of the vector space. Linear algebra explores vector spaces of finite dimension.

Between two vector spaces you can set the mapping . Linear algebra explores mappings, which are called linear . Linear mapping links between two vector spaces, constructed over one and the same field.

The main objects of a linear algebra are: arrays, vectors, and matrices.