## Linear programming - Wikipedia

The first four axioms mean that V is an abelian group under addition.. Elements of a vector space may have various nature; for example, they can be sequences, functions, polynomials or bia2clip.tk algebra is concerned with properties common to all vector spaces. Applications of Linear Algebra. Application 1: Constructing Curves and surfaces passing through Specified points. Jul 30, · You will therefore follow your movements, but also those of your friends if they decide to share their position with you. • Get ad free maps in Ondago. Ondago is an ever growing application: new maps are regularly added to our catalogue and new features are constantly worked on to keep enhancing your user experience/5(97).

## Linear algebra - Wikipedia

Linear algebra is the branch of mathematics concerning linear equations such as, **application lineaire**. Linear algebra is central to almost all areas of mathematics, **application lineaire**. For instance, linear algebra is fundamental in modern presentations of geometryincluding for defining basic objects such as linesplanes and rotations.

Also, functional analysis may be basically viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and engineering areas, because it allows modeling many natural phenomena, and efficiently computing with such models.

For nonlinear systemswhich cannot be modeled with linear algebra, linear algebra is often used as a first-order approximation. The procedure for solving simultaneous linear equations now called Gaussian elimination appears *application lineaire* the ancient Chinese mathematical text Chapter Eight: Rectangular Arrays of The Nine Chapters on the Mathematical Art.

Its use is illustrated in eighteen problems, with two to five equations. In fact, in this new geometry, now called Cartesian geometrylines **application lineaire** planes are represented by linear equations, and computing their intersections amounts to solving systems of linear equations.

The first systematic methods for solving linear systems used determinantsfirst considered by Leibniz in InGabriel Cramer used them for giving explicit solutions **application lineaire** linear systems, now called Cramer's rule. Later, Gauss further described the method of elimination, which was initially listed as an advancement in geodesy. In Hermann Grassmann published his "Theory of Extension" which included foundational new topics of what is today called linear algebra.

InJames Joseph Sylvester introduced the term matrixwhich is Latin for womb. Linear algebra grew with ideas noted in the complex plane. The segments are equipollent. Arthur Cayley introduced matrix multiplication and the inverse matrix inmaking possible the general linear *application lineaire.* The mechanism of group representation became available for describing complex and hypercomplex numbers. Crucially, Cayley used a single letter to denote a matrix, thus treating a matrix as an aggregate object.

He also realized the connection between matrices and determinants, and wrote "There would be many things to say about this theory of matrices *application lineaire* should, it seems to me, precede the theory of determinants". The telegraph required an explanatory system, **application lineaire**, and the publication of A Treatise on Electricity and Magnetism instituted a field theory of forces and required differential geometry for expression, **application lineaire**.

Linear algebra is flat differential geometry and serves in tangent spaces *application lineaire* manifolds. Electromagnetic symmetries of spacetime are expressed by the Lorentz transformationsand much of **application lineaire** history of linear algebra is the history of Lorentz transformations. The first modern and more precise definition of a vector space was introduced by Peano in ; [5] bya theory of linear transformations of finite-dimensional vector spaces had emerged.

Linear algebra took its modern form in the first half of the *application lineaire* century, when many ideas and methods of previous centuries were generalized as abstract algebra. The development of computers led to increased research in efficient algorithms for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modelling and simulations. Until the 19th century, linear algebra was introduced through systems of linear equations and matrices.

In modern mathematics, the presentation through vector spaces is generally preferred, since it is more synthetic, more general not limited to the finite-dimensional case*application lineaire*, and conceptually simpler, **application lineaire**, although more abstract. A vector space over a field F often the field of the real numbers is a set V equipped with two binary operations satisfying the following axioms.

Elements of V are called vectors*application lineaire*, *application lineaire* elements **application lineaire** F are called scalars. The second operation, scalar multiplicationtakes any scalar a and any vector v and outputs a new vector av, *application lineaire*. The axioms that addition and scalar multiplication must satisfy are the following.

In the list below, u *application lineaire,* v and w are arbitrary elements of V**application lineaire**, and a and b are arbitrary scalars in the field F.

The first four axioms mean that V is an abelian group under addition, *application lineaire*. Elements of a vector space may have various nature; for example, *application lineaire*, they can be sequencesfunctionspolynomials or **application lineaire.** Linear algebra is concerned with properties common **application lineaire** all vector spaces.

Linear maps are mappings between vector spaces that preserve the vector-space structure. Given two vector spaces V and W over a field Fa linear map also called, in **application lineaire** contexts, linear transformation, *application lineaire*, linear mapping or linear operator is a map, *application lineaire*. This implies that for any vectors uv in V and scalars ab in F*application lineaire*, one has, *application lineaire*.

When a bijective linear map exists between two vector spaces that is, every vector from the second space is associated with exactly one in the firstthe two **application lineaire** are isomorphic. Because an isomorphism preserves linear structure, two isomorphic vector spaces are "essentially the same" from the linear algebra point of view, in the sense that they cannot be distinguished by using vector space properties.

An *application lineaire* question in linear algebra is testing whether a linear map is an isomorphism or not, *application lineaire*, and, if it is not an isomorphism, finding its range or image and the set of elements that are mapped to the zero vector, called the kernel of the map. All these questions can be solved by using Gaussian elimination or some *application lineaire* of this algorithm.

The study of subsets of vector spaces that are themselves vector spaces for the induced operations is fundamental, similarly as for many mathematical structures. These subsets are called linear subspaces. These conditions suffice for implying that W is a vector space.

For example, the image of a linear map, and the inverse image of 0 by a linear map called kernel or null space are linear subspaces, **application lineaire**. Another important way of forming a subspace is to consider linear combinations of a set S of vectors: the set of all sums.

The span of S is also the intersection of all linear subspaces containing S. In other words, it is the smallest for the inclusion relation linear subspace containing S.

A set of vectors is linearly independent if none is in the span of the others. Equivalently, a set S of vector is linearly independent if the only way to express the zero vector as a linear combination of elements of S is to take zero for every coefficient a i. A set of vectors that spans a vector space is called a spanning set or generating set, *application lineaire*.

If a spanning set S is linearly dependent that is not linearly independentthen some element w of S is in the span of the other elements of Sand the span would remain the same if one remove w from S. One may continue to remove elements of S until getting a linearly independent spanning set. Such a linearly independent set that spans a vector space V is called a basis of V.

The importance of bases lies in the fact that there are together minimal generating sets and maximal independent sets. Any two bases of a vector space V have the same cardinalitywhich is called the dimension of V ; this is the dimension theorem for vector spaces, **application lineaire**. Moreover, two vector spaces over the same field F are isomorphic if and only if they have the same dimension. If any basis of V and therefore every basis has a finite number of elements, V is a finite-dimensional vector space.

Matrices allow explicit manipulation of finite-dimensional vector spaces and linear maps. Their theory is **application lineaire** an essential part of linear algebra, *application lineaire*. Let V be a finite-dimensional vector space over a field Fand v 1v 2By definition of a basis, the map. That is, if. Matrix multiplication is defined in such a way that the product of two matrices is the matrix of the composition of *application lineaire* corresponding linear maps, and the product of a matrix and a column matrix is the column matrix representing the result of applying the represented linear map to the represented vector, **application lineaire**.

*Application lineaire* follows that the theory of finite-dimensional vector spaces and the theory of matrices are two different languages for expressing exactly the same concepts. Two matrices that encode the same linear transformation in different bases are called similar.

Equivalently, two matrices are similar if one can transform one in the other by elementary row and column operations. For a matrix representing a linear map from W to V**application lineaire**, the row operations correspond to change of bases in V and the column operations correspond **application lineaire** change of bases in W, *application lineaire*.

Every matrix is similar to an identity matrix possibly bordered by zero rows and zero columns. In terms of vector space, this means that, for any linear map from W to V**application lineaire**, there are bases such that a part of the basis of W is mapped bijectively on a part of the basis of Vand that the remaining basis elements of Wif any, are mapped to zero this is a way of expressing the fundamental theorem of linear algebra. Gaussian elimination is the basic algorithm **application lineaire** finding these elementary operations, **application lineaire**, and proving this theorem.

Systems of linear equations form a fundamental part of linear algebra. Historically, linear algebra and matrix theory has been developed for solving such systems. In the modern presentation of linear algebra through vector spaces and matrices, many problems may be interpreted in terms of linear systems.

Let T be the linear **application lineaire** associated to the matrix M. A solution *application lineaire* the system S is a vector. Let S' be the associated homogeneous system*application lineaire*, where the right-hand sides of the equations are put to zero. The solutions of S' are exactly the elements of the kernel of T or, equivalently, M. The Gaussian-elimination consists of performing elementary row operations on the augmented matrix.

These row operations do not change the set of solutions of the system of equations. In the example, the reduced echelon form is, *application lineaire*. It follows from this matrix interpretation of linear systems that the same methods can be applied for solving linear systems and for many operations on matrices and linear transformations, which include the computation of the rankskernelsmatrix inverses.

A linear endomorphism is a linear map that maps a vector space V to itself. If V has a basis of n elements, such an endomorphism is represented by a square matrix of size n. With respect to general linear maps, linear endomorphisms and square matrices have some specific properties that make their study an important part of linear algebra, which is used in many parts of mathematics, including geometric transformationscoordinate changesquadratic formsand many other part of mathematics.

The determinant of a square matrix is a polynomial function of the entries of the matrix, such that the matrix is invertible if and only if the determinant is not zero.

This results from the fact that the determinant of a product of matrices is the product of the determinants, and thus that a matrix is invertible if and only if its **application lineaire** is invertible.

Cramer's rule is a closed-form expressionin terms of determinants, of the solution **application lineaire** a system of n linear equations in n unknowns. The determinant of an endomorphism is the determinant of the matrix representing the endomorphism in *application lineaire* of some ordered basis.

This definition makes sense, since this determinant is independent of the choice of the basis. This scalar a is an eigenvalue of f. If the dimension of V is finite, and a basis has been chosen, f and v may be represented, respectively, by a square matrix M and a column matrix z ; the equation defining eigenvectors and eigenvalues becomes.

Using the identity matrix Iwhose entries are all zero, except those of the main diagonal, which are equal to one, this may be rewritten.

The eigenvalues are thus the roots of the polynomial, **application lineaire**. If V is of dimension nthis is a monic polynomial of degree ncalled the characteristic polynomial of the matrix or of the endomorphismand there are, at most, n eigenvalues. If a basis exists that consists only of eigenvectors, the matrix of f on this basis has a very simple structure: it is a diagonal matrix such that the entries on the main diagonal are eigenvalues, and the other entries are zero, **application lineaire**.

In this case, the endomorphism and the matrix are said diagonalizable.

### Application linéaire — Wikipédia

application lin´eaire et que, munis de la norme uniforme, les espaces vectoriels V et C([0,1],R) sont des espaces de Banach, il r´esulte du th´eor`eme du graphe ferm´e que D est continu. En particulier, il existe une constante C > 0 telle que. Correction H [] Exercice 12 Soient E et F deux espaces vectoriels de dimension finie et ϕ une application linéaire de E dans F. Montrer que ϕ est un isomorphisme si et seulement si l’image par ϕ de toute base de E est une base de F. Correction H [] 4 Morphismes particuliers Exercice 13 Soit E l’espace vectoriel des. Linear programming (LP, also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization).