Theme 8. Vector operations in Maple

Linear algebra is a part of the algebra that studies the theory of linear equations, the theory of determinants, the theory of matrices, the theory of vector spaces and linear transformations, the theory of forms (quadratic), and the tensor calculus (partially).

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 in 1833 represented complex numbers in the form of a two-dimensional real vector space; he belongs to the authorship of the term "vector". The theory of matrices was developed in the works of Kelly (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. You can specify a map between two vector spaces . 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 the linear algebra are the following abstract concepts, such as arrays, vectors, and matrices.