Vector space

A vector space or vector space is an algebraic structure that is used in nearly every branch of mathematics. Vector spaces form the central object of study of linear algebra. The elements of a vector space are called vectors. They can be added or multiplied by scalars (numbers), the result is again a vector of the same vector space. The result is the term used by these characteristics were abstracted from starting vectors of the Euclidean space so that they are then transferred to more abstract objects such as functions or matrices.

The scalars, with which one can multiply a vector derived from a body. Therefore, a vector space is always a vector space over a particular body. Very often it involves the field of real numbers or the field of complex numbers. We then speak of a real or complex vector space.

A basis of a vector space is a set of vectors, which allows each vector by clear, finally represent many coordinates. The number of basis vectors is called a dimension of the vector space. It is independent of the choice of basis and can be infinite. The structural properties of a vector space are uniquely determined by the body on which it is defined and determined its dimension.

A base makes it possible to perform calculations with vectors over their coordinates instead of the vectors themselves, facilitating some applications.

• 7.1 Direct sum
• 7.2 Direct product
• 7.3 tensor

Definition

There are a lot of, a body, an internal binary operation called vector addition, and an external binary operation called scalar multiplication. It is then called a vector space over the field or short - vector space if for the vector addition of the properties

And on for the scalar properties

Are to all and fulfills.

Comments

• The axioms V1, V2 and V3 of the vector addition state that forms a group, and Axiom V4 that this is Abelian. Your identity element is called a null vector.
• A body is an abelian group with neutral element ( zero element ) and a second inner two -digit shortcut so also is an abelian group and the distributive laws apply. Important examples of bodies are the real numbers and the complex numbers.
• The axioms S1 and S2 of the scalar multiplication are also called distributive, Axiom S3 as associative law. However, please note that Axiom S2, the plus sign two different additions (left to those in and right in ), respectively, and that at Axiom S3, the scalar multiplication is associative with multiplication in.
• The axioms S1 and S2 ensure compatibility with the links of vector addition and the legal compatibility with the body and the vector addition of the scalar multiplication. Axioms S3 and S4 also ensure that the multiplicative group of the body operated on.
• In this article, both the addition in the body as well as the addition in the vector space are below, as is customary in mathematics, referred to with the same character, although there are different links. Multiplication in the same way both the body as well as the scalar multiplication between the body member and element can be designated by the vector space. In both multiplications, it is common to omit the Malpunkt. In practice, there is no fear that both the two additions or multiplications to be confused. The use of the same symbols makes the vector space axioms particularly suggestive.

First properties

For all and the following statements hold:

• .
• .
• The equation is uniquely solvable for all; the solution is.

Examples

Euclidean plane

A descriptive vector space is the two-dimensional Euclidean plane ( in rectangular Cartesian coordinates ) with the arrow classes ( displacements or translations) as vectors and the real numbers as scalars.

The sum of the two displacements is again a displacement, namely displacement of the one obtained by performing the two shifts in sequence:

The zero vector corresponds to the shift that leaves all the points in place, ie the identity map.

By the stretching of the displacement by a scalar of the set of real numbers we get three times the displacement:

More from this example applies as stated in the real affine plane.

Coordinate space

Is a body and a natural number, so the times Cartesian product forms

The set of all tuples with entries in a vector space over. The addition and scalar multiplication are defined componentwise; for you just put:

And

Often, the tuples are also listed as column vectors, ie their entries are written below each other. The vector spaces form as it were the standard examples of finite dimensional vector spaces. Every -dimensional vector space is isomorphic to the vector space. By means of a base of each element of a vector space can be uniquely represented by a member of a coordinate tuple.

Function spaces

If a body is a vector space and any amount as may, on the set of all functions of addition and scalar multiplication are defined pointwise: For and, the functions and defined by

With this addition and scalar multiplication is a vector space. In particular, this applies to, so if the target space of the body itself is selected. Other examples of vector spaces obtained as subspaces of function spaces.

Space of linear functions

A simple example of a two-dimensional feature space is the space of real -linear functions, that is, the functions of the form

With real numbers and. These are functions whose graph is a straight line. The amount of these functions is a subspace of the space of all the real functions, because the sum of two linear functions is again linear and is a multiple of a linear function is a linear function.

For example, the sum of the two linear functions and

The function with

3 times of the linear function is the linear function

Polynomräume

The set of polynomials with coefficients in a body forms, with the usual addition of the usual multiplication, and with a body member, an infinite-dimensional vector space. The set of monomials is a basis of this vector space. The set of polynomials whose degree is bounded by an upward, forming a sub-vector space of dimension. For example, the set of all polynomials of degree less than or equal to 4, ie all polynomials of the form

A 5- dimensional vector space with the base.

With infinite bodies, one can identify the ( abstract ) polynomials with the corresponding polynomial. From this perspective, the Polynomräume subspaces of the space of all functions from correct to. For example, the space of all real polynomials of degree equal to the space of linear functions.

Field extensions

If an upper body, then with his addition and multiplication as the restricted scalar multiplication of a vector space. The rules to be detected arising directly from the body of axioms. This observation is important in the field theory.

For example, in this way a two-dimensional vector space; is a base. Similarly, a infinite-dimensional vector space, but in which a base can not be specified in concrete terms.

Linear maps

Linear mappings are the functions between two vector spaces that preserve the structure of the vector space in a certain sense, they are the homomorphisms between vector spaces in the sense of universal algebra. A function between two vector spaces over the same body and is said to be linear if for all and all

Are fulfilled. That is compatible with the structures that constitute the vector space, namely, the addition and of the scalar multiplication in a certain sense. Two vector spaces are called isomorphic if there is a linear map between them, which is bijective, ie has an inverse function. This inverse function is then automatically also linear. Isomorphic vector spaces do not differ with respect to their structure as a vector space.

Basis of a vector space

For a finite number and is defined as the sum

As a linear combination of the vectors. This itself is again a vector of the vector space.

Is a subset of, so is the set of all linear combinations of vectors of the linear hull of being called. It is a subspace of, namely the smallest subspace that contains.

A subset of a vector space is called linearly dependent if the zero vector can be expressed in a non - trivial way as a linear combination of vectors. "Non- trivial " means that at least a scalar ( a coefficient of the linear combination ) is different from zero. Otherwise, ie linearly independent.

A subset of a vector space is a basis of if is linearly independent and the linear hull of the whole vector space.

Assuming the axiom of choice can be set via the lemma of Zorn prove that every vector space has a basis ( it is free ), this statement in the context of Zermelo Fraenkel equivalent to the axiom of choice is. This has far -reaching consequences for the structure of each vector space: First of all, it can be shown that any two bases of a vector space have the same cardinality, so that the cardinality of any basis of a vector space is a unique cardinal number that is called the dimension of the vector space. Two vector spaces over the same body are now isomorphic if they have the same dimension, as a result of equal thickness of two bases of two vector spaces, there is a bijection between them. This can be, continue to a bijective linear mapping, ie a isomorphism of the two vector spaces. Similarly it can be shown that any linear mappings are defined by the images of elements of a basis. This allows the representation of any linear maps between finite dimensional vector spaces as a matrix. This can be extended to infinite-dimensional vector spaces, but it must be ensured that any generalized " column" contains only finitely many non-zero entries so that each basis vector is mapped to a linear combination of basis vectors in the target area.

Using the basic concept has the problem of finding a skeleton in the category of all vector spaces over a given body, reduced it to find a skeleton in the category of sets, which is given by the class of cardinal numbers. Each one -dimensional vector space can also be regarded as the -fold direct sum of the underlying body. The direct sums of a body thus form a skeleton of the category of vector spaces over him.

The linear factors for the representation of a vector in the basis vectors called the coordinate vector with respect to the base and the elements of the underlying body. Only by introducing a basis be assigned its coordinates with respect to the chosen basis of each vector. Thus, the calculation is made easier, especially if you can use their assigned " ideological " coordinate vectors rather than vectors in " abstract " vector spaces.

Subspace

A subspace (also linear subspace ) is a subset of a vector space that is itself a vector space over the same body. The vector space operations are inherited to the sub-vector space. Is a vector space over a field, then a subset is exactly then a sub-vector space if the following conditions are met:

• Applies to all
• For all and is

The amount must therefore closed under vector addition and scalar multiplication be. Every vector space contains two trivial subspaces, one being himself, on the other hand the zero vector space, which consists only of the zero vector. Each subspace is an image of another vector space with a linear mapping in the space and kernel of a linear mapping into another vector space. Of a vector space, and a subspace can be a further vector space, the space ratio or factor space form, which is related with the significant sub-space of a feature to be a core, see also homomorphism by forming equivalence classes.

Combination of vector spaces

Two or more vector spaces can be linked together in different ways, so that a new vector space is created.

Direct sum

The direct sum of the two vector spaces with the same body consists of all ordered pairs of vectors, from which originates the first component from the first space and the second component from the second space:

On this set of pairs vector addition and scalar multiplication is then defined componentwise, which in turn results in a vector space. The dimension is then equal to the sum of the dimensions of and. Often the elements of place are written as a pair as a total. The direct sum can be generalized to the sum of finitely many or even an infinite number of vector spaces, in the latter case only finitely many components must be equal to the zero vector.

Direct product

The direct product of two vector spaces over the body is the same as the direct sum of all ordered pairs of vectors of the form

The vector and the scalar addition is component-wise again and the defined dimension is again equal to the sum of the dimensions of and. In the direct product of infinitely many vector spaces but many components must also be infinite equal to the zero vector, which distinguishes it in this case of the direct sum.

Tensor

The tensor product of two vector spaces over the same body is

Noted. The elements of the Tensorproduktraums have here the bilinear representation

Which are scalars, is a basis of, and is a basis of. Is or infinite-dimensional finitely many summands may in this case be equal to zero again. The dimension of the product is then equal to the dimensions of and. Also, the tensor can be generalized to multiple vector spaces.

Vector spaces with additional structure

In many applications in mathematics, such as the geometry or analysis, the structure of a vector space is insufficient, such permit vector spaces to be no limit processes, and therefore we consider vector spaces with certain additionally defined on them structures that are compatible with the vector space structure in certain senses are. Examples:

In all these examples are topological vector spaces. In topological vector spaces, the analytical concepts of convergence, uniform convergence and completeness are applicable. A complete normed vector space is called a Banach space, a complete pre-Hilbert space is called Hilbert space.

Generalizations

• If a commutative ring one assumes instead a body, you get a module. Modules are a common generalization of the concepts of abelian group (for the ring of integers ) and vector space ( for the body).

• Some authors do not go into the definition of bodies on the commutative property of multiplication and call modules over skew fields also vector spaces. If you follow this procedure, so law and vector spaces Links vector spaces must be distinguished when the skew field is commutative. The above definition of the vector space here results in a left - vector space, since the scalars are on the left in the product. Right - vector spaces are defined analogously with the mirror image declared scalar multiplication. Many fundamental results are completely analogous for vector spaces over skew fields, such as the existence of a base.

• If one assumes half body instead of a body, you get a half- vector space.

• Another generalization of vector spaces are vector bundles; they consist of a vector space for each point of a topological base space.