# Vector space

 Vector space affects the special fields mathematics is special case of Abel group carries operation of a body module enclosure as special cases body (VR over itself) topological vector space standardized area algebra (with internal multiplication)

a vector space is a mathematical structure (also structure algebraic in particular), which is used in nearly all branches of mathematics. In detail to be regarded vector spaces in that Linear algebra.

A vector space consists of individual vectors, which can be added or be multiplied by a scalar number, so that the result is in each case vector of the same vector space. The elements of a vector space cannot only outvectors admitted of geometry its, but also abstraktere objects such as functions or stencils.

Since the scalar numbers, by which one can multiply a vector, a body to come of, is a vector space always a vector space „over “a certain body. One speaksfor example of a vector space over the real numbers. In most applications one takes these or the complex numbers as a basis.

A basis of a vector space is a quantity of vectors, it permits each vector by clear coordinates to describe.Thus counting in vector spaces is facilitated. The number of basis vectors is called dimension of the vector space and is a measure for its size.

## formal definition

a vector space over a body$K< /math> or K-vector space is briefly a abelsche group [itex] (V, +, 0)< /math> together with one as scalar multiplication designated linkage < math> \ cdot: K \ times V \ tons of V< /math> The scalar multiplication must thereby for all [itex] u, v \ in V< /math> and [itex] \ alpha, \ beta \ in K< /math> the following conditions fulfill: Assoziativität < math> \ alpha \ cdot (\ beta\ cdot v) = (\ alpha \ cdot \ beta) \ cdot v< /math> Distributive laws < math> \ alpha \ cdot (u + v) = \ alpha \ cdot u + \ alpha \ v /math< cdot> [itex] (\ alpha + \ beta) \ cdot v = \ alpha \ cdot v + \ beta \ cdot v< /math> Neutrality of the 1 of the body[itex] K< /math> [itex] 1 \ cdot v = v$

A K-vector space nothing else than a k link module , its basic mounting ring is differently expressed $K< /math> even a body is. ## notes • the addition of the abelschen group [itex] (V, +, 0)< /math> Vektoraddition is called, the neutral element [itex] 0< /math> the zero-vector. • The distributive laws guarantee the compatibility of Vektoraddition and scalar multiplication. • Although the multiplication in the body [itex] K< /math> and the scalar multiplication to be confounded, they may not become usually both with the same indication „[itex] \ cdot< /math> “designation. Oftenone omits the multiplication symbol even completely. ## first characteristics for all [itex] \ alpha \ in K< /math> and [itex] v \ in V< /math> applies: • [itex] (- \ alpha) \ cdot v = - (\ alpha \ cdot v) = \ alpha \ cdot (- v)< /math>. • [itex] \ alpha \ cdot v = 0 \ quad \ Leftrightarrow \ quad \ alpha=0< /math> or [itex] v = 0< /math>. The equation [itex] v+x =w< /math> v is, <w> \ in V /math for all< math> clearly solvable; the solution is [itex] x = w + (- v)< /math>. ## examples ### Euclidean level a descriptive vector space is the two-dimensional Euclidean level [itex] \ R^2< /math> with the arrow classes (shifts) as vectors and the real numbers as scalars. [itex] \ mathbf {v} = (2, 3)$ the shift is around 2 units to the right and 3 units upward,
< math> \ mathbf {w} =(3, - 5) [/itex] the shift is around 3 units to the right and 5 units downward.

The sum of two shifts is again a shift:

$\ mathbf {v} + \ mathbf {w} = (5, - 2)< /math>, i.e. 5 units to the rightand 2 units downward. The zero-vector [itex] \ mathbf {0} = (0, 0)$ corresponds to no shift.

By the aspect ratio of the shift $\ mathbf {v}< /math> with a scalar [itex] A = 3< /math> from the quantity of the real numbers we receive the three-waythe shift: [itex] A \ cdot \ mathbf {v} = 3 \ cdot (2, 3) = (6, 9)< /math>. ### area of the affinen functions already somewhat a abstrakterer vector space is the area of the affinen functions on the realNumbers. These are the functions of the form < math> f:\R\to\R, \; x \ mapsto A \ cdot x + b< /math> with real numbers [itex] A< /math> and [itex] b< /math>. This all functions, whose graph is a straight line, are descriptive spoken. In this opinion our area produces all straight lines up to thoseexactly perpendicularly standing. We select exemplarily two linear functions < math> f (x) = 2x + 3< /math> , [itex] = 3x - 5 /math, then we <see> g (x), how their sum results in again a affine function: [itex] f (x) + g (x) = 2x +3 + 3x - 5 = (2+3) x + (3-5) = 5x - 2$

The zero-vector is the constant function

< math> 0 = 0x + 0 [/itex], which illustrate all points on the zero.

With a scalar $A = 3< /math>from the quantity of the real numbers the scalar multiplication results in < math> A \ cdot f (x) = 3 \ cdot (2x + 3) = (3 \ cdot 2) x + (3 \ cdot 3) = 6x + 9$.

## Verallgemeinerungen

• if one in place of a body [itex] K< /math> at the basis one puts, receives a ring a module. Modules are a common Verallgemeinerung of the terms abelsche group (for the ringthe whole numbers) and vector space (for bodies).
• Some authors do in the definition of bodies without the commutative law of the multiplication and call likewise modules over inclined bodies vector spaces. If one follows this proceeding, then clinching vector spaces and K-Rechtsvektorräume must be differentiated, ifthe inclined body is not commutative. The definition of the vector space given above results in in the case of a clinching vector space since the scalars in the product on the left side. K-Rechtsvektorräume are defined similarly with the mirror-image explained scalar multiplication.
• If one in place of a body[itex] K< /math> at the basis one puts, receives a half body a half vector space.