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Given a vector space V over a field K, the span of a set S ( not necessarily finite ) is defined to be the intersection W of all subspaces of V which contain S. W is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W.

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## Some Related Sentences

Given and vector

__Given__any

__vector__

**space**

**V**

**over**

**a**

**field**F

**,**

**the**dual

**space**

**V***

**is**

**defined**

**as**

**the**

**set**

**of**

**all**linear maps

**(**linear functionals ).

__Given__

**a**basis

**of**

**a**

__vector__

**space**

**,**every element

**of**

**the**

__vector__

**space**can

**be**expressed uniquely

**as**

**a**

**finite**linear combination

**of**basis

**vectors**

**.**

__Given__

**a**

__vector__

**space**

**V**and

**a**quadratic form g an explicit matrix representation

**of**

**the**Clifford algebra can

**be**

**defined**

**as**follows

**.**

__Given__

**a**

__vector__v

**in**R < sup > n </ sup > one defines

**the**directional derivative

**of**

**a**smooth map ƒ: R < sup > n </ sup >→ R at

**a**point x

**by**

__Given__any

__vector__

**space**

**V**

**over**

**K**we can construct

**the**tensor algebra T

**(**

**V**

**)**

**of**

**V**

**.**The tensor algebra

**is**characterized

**by**

**the**fact:

__Given__that this

**is**

**a**plane wave

**,**each

__vector__represents

**the**magnitude and direction

**of**

**the**electric

**field**for an entire plane that

**is**perpendicular

**to**

**the**axis

**.**

__Given__

**a**normalized light

__vector__l

**(**pointing from

**the**light source toward

**the**surface

**)**and

**a**normalized plane normal

__vector__n

**,**one can work out

**the**normalized reflected and refracted rays:

__Given__

**the**dimensions

**of**

**the**ellipsoid

**,**

**the**conversion from lat / lon / height-above-ellipsoid coordinates

**to**X-Y-Z

**is**straightforward — calculate

**the**X-Y-Z for

**the**given lat-lon on

**the**surface

**of**

**the**ellipsoid and add

**the**X-Y-Z

__vector__that

**is**perpendicular

**to**

**the**ellipsoid there and has length equal

**to**

**the**point's height above

**the**ellipsoid

**.**

__Given__

**a**subset

**S**

**in**R < sup > n </ sup >,

**a**

__vector__

**field**

**is**represented

**by**

**a**vector-valued function

**V**:

**S**→ R < sup > n </ sup >

**in**standard Cartesian coordinates

**(**x < sub > 1 </ sub >, ..., x < sub > n </ sub >).

__Given__two C < sup > k </ sup >-

__vector__fields

**V**

**,**

**W**

**defined**on

**S**and

**a**real valued C < sup > k </ sup >- function f

**defined**on

**S**

**,**

**the**two operations scalar multiplication and

__vector__addition

__Given__

**a**differentiable manifold M

**,**

**a**

__vector__

**field**on M

**is**an assignment

**of**

**a**tangent

__vector__

**to**each point

**in**M

**.**More precisely

**,**

**a**

__vector__

**field**F

**is**

**a**mapping from M into

**the**tangent bundle TM so that

**is**

**the**identity mapping

__Given__

**a**

__vector__x ∈

**V**and y * ∈

**W***, then

**the**tensor product y * ⊗ x corresponds

**to**

**the**map A:

**W**→

**V**given

**by**

__Given__ω =

**(**xθ

**,**yθ

**,**zθ ), with v =

**(**x

**,**y

**,**z

**)**

**a**unit

__vector__

**,**

**the**correct skew-symmetric matrix form

**of**ω

**is**

__Given__two column

**vectors**

**,**their dot product can also

**be**obtained

**by**multiplying

**the**transpose

**of**one

__vector__with

**the**other

__vector__and extracting

**the**unique coefficient

**of**

**the**resulting 1 × 1 matrix

**.**

Given and space

*

__Given__any Banach__space__X**,****the**continuous linear operators A: X → X form**a**unitary associative algebra**(**using composition**of**operators**as**multiplication ); this**is****a**Banach algebra**.**
*

__Given__any topological__space__X**,****the**continuous real-or complex-valued functions on X form**a**real**or**complex unitary associative algebra ; here**the**functions are added and multiplied pointwise**.**__Given__

**a**

**finite**dimensional real quadratic

__space__with quadratic form

**,**

**the**geometric algebra for this quadratic

__space__

**is**

**the**Clifford algebra Cℓ

**(**

**V**

**,**Q ).

__Given__these two assumptions

**,**

**the**coordinates

**of**

**the**same event

**(**

**a**point

**in**

__space__and time

**)**described

**in**two inertial reference frames are related

**by**

**a**Galilean transformation

**.**

__Given__infinite

__space__

**,**there would

**,**

**in**fact

**,**

**be**an infinite number

**of**Hubble volumes identical

**to**ours

**in**

**the**universe

**.**

__Given__

**a**

**set**

**of**training examples

**of**

**the**form

**,**

**a**learning algorithm seeks

**a**function

**,**where

**is**

**the**input

__space__and

__Given__

**a**point x

**in**

**a**topological

__space__

**,**let N < sub > x </ sub > denote

**the**

**set**

**of**

**all**neighbourhoods containing x

**.**

__Given__

**the**date

**of**his publication and

**the**widespread

**,**permanent distribution

**of**his work

**,**it appears that he should

**be**regarded

**as**

**the**originator

**of**

**the**concept

**of**

__space__sailing

**by**light pressure

**,**although he did

**not**develop

**the**concept further

**.**

__Given__an arbitrary topological

__space__

**(**X

**,**τ

**)**there

**is**

**a**universal way

**of**associating

**a**completely regular

__space__with

**(**X

**,**τ ).

__Given__any embedding

**of**

**a**Tychonoff

__space__X

**in**

**a**compact Hausdorff

__space__

**K**

**the**closure

**of**

**the**image

**of**X

**in**

**K**

**is**

**a**compactification

**of**X

**.**

__Given__

**a**completely regular

__space__X there

**is**usually more than one uniformity on X that

**is**compatible with

**the**topology

**of**X

**.**

__Given__

**the**

__space__X = Spec

**(**R

**)**with

**the**Zariski topology

**,**

**the**structure sheaf O < sub > X </ sub >

**is**

**defined**on

**the**D < sub > f </ sub >

**by**setting Γ

**(**D < sub > f </ sub >, O < sub > X </ sub >) = R < sub > f </ sub >,

**the**localization

**of**R at

**the**multiplicative system

Given and V

__Given__

**a**

**set**

**S**with

**a**partial order ≤, an infinite descending chain

**is**

**a**chain

__V__that

**is**

**a**subset

**of**

**S**upon

**which**≤ defines

**a**total order such that

__V__has no least element

**,**that

**is**

**,**an element m such that for

**all**elements n

**in**

__V__it holds that m ≤ n

**.**

__Given__

**a**class function G:

__V__→

__V__

**,**there exists

**a**unique transfinite sequence F: Ord →

__V__

**(**where Ord

**is**

**the**class

**of**

**all**ordinals

**)**such that

__Given__

**a**topological

**space**X

**,**denote F

**the**

**set**

**of**filters on X

**,**x ∈ X

**a**point

**,**

__V__

**(**x

**)**∈ F

**the**neighborhood filter

**of**x

**,**A ∈ F

**a**particular filter and

**the**

**set**

**of**filters finer than A and that converge

**to**x

**.**

A Clifford algebra Cℓ

**(**__V__**,**Q**)****is****a**unital associative algebra**over****K**together with**a**linear map satisfying for**all****defined****by****the**following universal property:__Given__any associative algebra A**over****K**and any linear map such that

Given and over

"

__Given__its general release**in**January 1931**,**City Lights proved**to****be****a**popular and financial success — eventually grossing__over__$ 5 million**.**__Given__

**the**large number

**of**U

**.**

**S**

**.**military personnel and their dependents residing

**in**Europe

**,**it was expected that

__over__7 %

**of**donors would

**be**deferred due

**to**

**the**policy

**.**

__Given__that

**the**ColecoVision could produce near arcade-quality ports

**,**industry magazines like Electronic Games were unanimous

**in**their enthusiasm

__over__

**the**console

**.**

__Given__that

**the**dispute with Pizarro

__over__Cuzco had kept intensifying

**,**Almagro spent

**a**great deal

**of**time and money equipping

**a**company

**of**500 men for

**a**new exploration south

**of**Peru

**.**

*

__Given__**a**transformation group G__over__A**,**there exists an equivalence relation ~__over__A**,**whose equivalence classes are**the**orbits**of**G**.**__Given__these origins

**,**there have been various suggestions

__over__

**the**years

**to**rename

**the**town

**(**for example

**,**

**to**" Invernevis ").

__Given__

**a**

**field**

**K**

**,**

**the**corresponding general linear groupoid GL < sub >*</ sub >(

**K**

**)**consists

**of**

**all**invertible matrices whose entries range

__over__

**K**

**.**Matrix multiplication interprets composition

**.**

__Given__

**the**differences

**of**opinion within

**the**Lord Aberdeen cabinet

__over__

**the**direction

**of**foreign policy with regard

**to**relations between Britain and

**the**French under Napoleon III

**,**it

**is**

**not**surprising that debate raged within

**the**government

**as**Louis Bonaparte

**,**now assuming

**the**title

**of**Emperor Napoleon III

**of**France

**.**

On poverty

**,**Hoover said that "__Given__**the**chance**to**go forward with**the**policies**of****the**last eight years**,**we shall soon with**the**help**of**God**,****be****in**sight**of****the**day when poverty will**be**banished from this nation ", and promised**,**" We**in**America today are nearer**to****the**final triumph__over__poverty than ever before**in****the**history**of**any land ," but within months**,****the**Stock Market Crash**of**1929 occurred**,**and**the**world's economy spiraled downward into**the**Great Depression**.**__Given__

**the**extraordinary competition and technological craftsmanship required for top search engine placement

**,**

**the**implication

**of**

**the**term " search engine optimization " has deteriorated

__over__

**the**last decade

**.**

__Given__that

**the**cost

**of**replacing an executive can run

__over__100 %

**of**his

**or**her annual salary

**,**any investment

**of**time and energy

**in**re-recruitment will likely pay for itself many times

__over__if it helps

**a**business retain just

**a**handful

**of**key players that would have otherwise left

**.**

__Given__

**a**function ƒ

**defined**

__over__

**the**reals x

**,**and its derivative ƒ < nowiki > '</ nowiki >, we begin with

**a**first guess x < sub > 0 </ sub > for

**a**root

**of**

**the**function f

**.**Provided

**the**function

**is**reasonably well-behaved

**a**better approximation x < sub > 1 </ sub >

**is**

The Fascist government had direct rule only

__over__**the**Benadir territory__Given__**the**defeat**of****the**Dervish movement**in****the**early 1920s and**the**rise**of**fascism**in**Europe**,**on 10 July 1925**,**Mussolini gave**the**green light**to**De Vecchi**to**start**the**takeover**of****the**north-eastern sultanates**.**__Given__

**a**particular wing and its velocity through

**the**air

**,**debates

__over__

**which**mathematical approach

**is**

**the**most convenient

**to**use can

**be**misperceived

**by**novices

**as**differences

**of**opinion about

**the**basic principles

**of**flight

**.**

"

__Given__their 3: 00**a****.**m**.**start time and non-inclusion**(**at frontman John Fogerty's insistence**)****in****the**Woodstock film**,**members have expressed bitterness__over__their experiences at**the**famed festival**.**__Given__two affine spaces and

**,**

__over__

**the**same

**field**

**,**

**a**function

**is**an affine map if and only if for every family

**of**weighted points

**in**such that

__Given__that radio communication channels are allocated

**by**agencies such

**as**

**the**Federal Communication Commission giving

**a**prescribed

**(**maximum

**)**bandwidth

**,**

**the**advantage

**of**QPSK

__over__BPSK becomes evident: QPSK transmits twice

**the**data rate

**in**

**a**given bandwidth compared

**to**BPSK-at

**the**same BER

**.**

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