Linear algebra

The linear algebra is the branch of the Mathématiques which is interested in the study of the vector spaces (or linear spaces), their elements the Vecteur S, linear transformations and linear systems of equations (theory of the matrix S).

History

The history of the linear algebra starts with Rene Descartes which it first installation of the problems of Géométrie, like the intersection of two right, in the form of linear equation. It then establishes a bridge between two mathematical branches until now separate: algebra and geometry. If it does not define the basic notion of the linear algebra which is the vector space, it uses it already successfully. After this discovery progress in linear algebra will be limited to specific studies like the definition and the analysis of the first properties of the determinant S by Jean d' Alembert.

It is only at the 19th century that the linear algebra becomes a branch of mathematics with whole share. Carl Friedrich Gauss finds a method generic for the resolution of the linear systems of equations, Marie Ennemond Camille Jordan definitively solves the problem of the Réduction of endomorphism. In 1843, William Rowan Hamilton (inventive of the term vector ) discovers the Quaternion S. In 1844, Hermann Grassmann publishes a book Die lineare Ausdehnungslehre .

The beginning of the 20th century sees the birth of the modern formalization of mathematics. The vector spaces then become an omnipresent general structure in almost all the mathematical fields.

Interest

In their simplest form the vector spaces intuitively represent displacements in elementary geometrical spaces like the right , the plan or our physical space. The bases of this theory now replace the representation built by Euclide at third century BC. Modern construction makes it possible to generalize the concept of space to unspecified dimensions.

The linear algebra makes it possible to solve a whole whole of equations known as linear used not only in mathematics or Mécanique, but in many other branches like the Natural science or the Social sciences.

The vector spaces form also a fundamental tool for the Engineerings and are used as a basis for many fields in the Operations research.

This branch provides as an important theoretical support in Informatique, as it is material with calculators or array processors or software. A data-processing language left as of 1969 adopted generalized notations of the linear algebra: the language APL.

Lastly, it is a tool used in mathematics to solve problems as various as the Théorie of the groups, of the rings or of the bodies, the analyzes functional, the differential Géométrie or the Théorie of the numbers.

Elementary presentation

The linear algebra starts with the study of Vecteur S in Cartesian spaces of dimension 2 and 3. A vector, here, is a segment of right-hand side characterized at the same time by its length (or standard ), its direction and its direction. The vectors can then be used to represent certain physical entities like displacements, added or multiplied by scalars ( numbers), thus forming the first concrete example of vector Space.

The modern linear algebra was extended to consider spaces of arbitrary or infinite size. A vector space of dimension N is called a N-space. The majority of the results obtained in 2-spaces and 3-spaces can be wide with spaces of higher size. Although many people cannot correctly apprehend a vector in a N-space, they are useful to represent data. The vectors being ordered lists with N components, one can effectively handle these data in this environment. For example in economy, one can create and use vectors with eight dimensions to represent the rough National product of eight country.

Some theorems

  • Any space vector of finished size has at least a bases.
  • All the bases of the same vector space of finished size have even many vectors
  • Théorème of the “incomplete base”: either E a vector space of finished size, G a generating family of E and L a free family of vectors of G. Then there exists at least a base B of E such as L or included in B and B included in G.
  • Any space vector has a dual Espace A*; if has is of finished size, A* is of the same dimension.
  • Formula of Grassmann: Are E and F two pennies vector spaces of the same vector space of finished size. One has then:

Dim (E + G) = Dim (E) + Dim (G) - Dim (E ∩ G)

Other theorems relate to the conditions of inversion of matrix S of various types:

  • diagonal Matrix
  • triangular band
  • Matrix
  • with dominant diagonal (very much used in numerical analysis)

A theorem interesting at the time of the memories of computers of small size was that one could work separately on subsets (“blocks”) of a matrix by then combining them by the same rules that one uses to combine scalars in the matrices. With the current memories of several Gigaoctet S, this question lost a little its practical interest, but remains very appraisal in Théorie of the numbers, for the Décomposition in product of factors first with the Crible lieutenant-general of numbers (GNFS) ( Lanczos method per blocks ).

See too

Internal bonds

External bonds

  • Linear Algebra by Elmer G. Wiens
  • courses of the ROSO, of which Linear algebra
  • Ember: the base reasoned of exercises of mathematics and its chapter on the Linear algebra

Simple: Linear will algebra

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