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Baixe grĂ¡tis o arquivo Linear Algebra and Geometry Kostrikin&terney.info enviado por Volume 1 Linear Algebra and Geometry. A. I. Kostrikin and Yu. 1. Manin. LINEAR ALGEBRA AND. GEOMETRY. Paperback Edition. Alexei I. Kostrikin. Moscow State University, Russia and. Yuri I. Manin. Max-Planck Institut fur. Linear Algebra and Geometry Algebra Logic and Applications - Ebook download as PDF Algebra I Basic Notions Of Algebra - Kostrikin A I, Shafarevich I terney.info

MA Gel'fand, I. Van Nostrand Company, Inc. Linear Spaces 1. Vectors, whose starting points are located at a fixed point in space, can be multiplied by a number and added by the parallelogram rule. This is the classical model of the laws of addition of displacements, velocities, and forces in mechanics. In the general definition of a vector or a linear space, the real numbers are replaced by an arbitrary field and the simplest properties of addition and multiplication of vectors are postulated as an axiom. No traces of the "three-dimensionality" of physical space remain in the definition. The concept of dimensionality is introduced and studied separately. Analytic geometry in two- and three-dimensional space furnishes many exam- ples of the geometric interpretation of algebraic relations between two or three variables. However, as expressed by N. Bourbaki, " Definition A set is said to be a linear or vector space L over a field K if it is equipped with a binary operation L x L - L, usually denoted as addition 11i12 -. L, usually denoted as multiplication a, 1 - al, which satisfy the following axioms: a Addition of the elements of L, or vectors, transforms L into a commutative abelian group. Its zero element is usually denoted by 0; the element inverse to 1 is usually denoted by MANIN 1.

Analogously, the expression ala The following examples of linear spaces will be encountered often in what follows.

Zero-dimensional space. Caution: zero- dimensional spaces over different fields are different spaces: the field K is specified in the definition of the linear space. The basic field K as a one-dimensional coordinate space. The validity of the axioms of the linear space follows from the axioms of the field.

More generally, for any field K and a subfield K of it, K can be interpreted as a linear space over K.

For example, the field of complex numbers C is a linear space over the field of real numbers R, which in its turn is a linear space over the field of rational numbers Q. The elements of L can be written in the form of rows of length n a,, Addition and multiplication by a scalar is defined by the formulas: a,, One-dimensional spaces over K are called straight lines or K-lines; two-dimensional spaces are called K-planes.

Function spaces.

As usual, if f : S - K is such a function, then f s denotes the value off on the element s E S. The addition and multiplication rules are consistent with respect to this identification. Indeed, this equality follows from the fact that the left side equals the right side at every point s E S.

If the set S is infinite, then this result is incorrect. More precisely, it cannot be formulated on the basis of our definitions: sums of an infinite number of vectors in a general linear space are not defined!

Some infinite sums can be defined in linear spaces which are equipped with the concept of a limit or a topology see Chapter Such spaces form the basic subject of functional analysis. Linear conditions and linear subspaces. In analysis, primarily realvalued functions defined over all R or on intervals a, b C R are studied. For most applications, however, the space of all such functions is too large: it is useful to study continuous or differentiable functions. After the appropriate definitions are introduced, it is usually proved that the sum of continuous functions is continuous and that the product of a continuous function by a scalar is continuous; the same assertions are also proved for differentiability.

This means that the continuous or differentiable functions themselves form a linear space. Then M together with the operations induced by the operations in L in other words, the restrictions of the operations defined in L to M is called a linear subspace of L, and the conditions which an arbitrary vector in L must satisfy in order to belong to M are called linear conditions.

Here is an example of linear conditions in the coordinate space Kn.

We fix scalars a,, In other words, the intersection of any number of linear subspaces is also a linear subspace check this! We shall prove later that any subspace in K" is described by a finite number of conditions of the form 1.

An important example of a linear condition is the following construction. The dual linear space. Let L be a linear space over K. We shall first study the linear space F L of all functions on L with values in K. We assert that linear functions form a linear subspace of F L or "the condition of linearity is a linear condition".

The space of linear functions on a linear space L is called a dual space or the space conjugate to L and is denoted by L. In what follows we shall encounter many other constructions of linear spaces. Remarks regarding notation. It is very convenient, but not entirely consistent, to denote the zero element and addition in K and L by the same symbols.

Here are two examples of cases when a different notation is preferable. We regard L as an abelian group with respect to multiplication and we introduce in L multiplication by a scalar from R according to the formula a, z za. It is easy to verify that all conditions of Definition 1.

We define a new vector space L with the same additive group L, but a different law of multiplication by a scalar: a, l i-4 al, where a is the complex conjugate of a.

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Differing from existing textbooks in approach, the work illustrates the many-sided applications and connections of linear algebra with functional analysis, quantum mechanics and algebraic and differential geometry. The subjects covered in some detail include normed linear spaces, functions of linear operators, the basic structures of quantum mechanics and an introduction to linear programming. Also discussed are Kahler's metic, the theory of Hilbert polynomials, and projective and affine geometries.

Unusual in its extensive use of applications in physics to clarify each topic, this comprehensice volume should be of particular interest to advanced undergraduates and graduates in mathematics and physics, and to lecturers in linear and multilinear algebra, linear programming and quantum mechanics.