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In the [[mathematics|mathematical]] subfield of [[linear algebra]] or more generally [[functional analysis]], the '''linear span''' (also called the '''linear hull''') of a [[Set (mathematics)|set]] of [[vector space|vectors]] in a [[vector space]] is the [[intersection (set theory)|intersection]] of all [[Linear subspace|subspaces]] containing that set. The linear span of a set of vectors is therefore a vector space.
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==Definition==
 
Given a [[vector space]] ''V'' over a [[field (mathematics)|field]] ''K'', the span of a [[Set (mathematics)|set]] ''S'' of vectors (not necessarily finite) is defined to be the intersection ''W'' of all [[linear subspace|subspaces]] of ''V'' that 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'', and we say that ''S'' ''spans'' ''W''.
 
Alternatively, the span of ''S'' may be defined as the set of all finite [[linear combination]]s of elements of ''S'', which follows from the above definition.
 
:<math>\operatorname{span}(S) =  \left \{ {\sum_{i=1}^k \lambda_i v_i \mid k \in \mathbb{N}, v_i  \in S, \lambda _i  \in \mathbf{K}} \right \}.</math>
 
In particular, if ''S'' is a [[finite set|finite]] subset of ''V'', then the span of ''S'' is the set of all linear combinations of the elements of ''S''. In the case of infinite ''S'', infinite linear combinations (i.e. where a combination may involve an infinite sum, assuming such sums are defined somehow, e.g. if ''V'' is a [[Banach space]]) are excluded by the definition; a [[Linear combination#Generalizations|generalization]] that allows these is not equivalent.
 
== Examples ==
 
The [[real number|real]] vector space '''R'''<sup>3</sup> has {(2,0,0), (0,1,0), (0,0,1)} as a spanning set. This particular spanning set is also a [[Basis (linear algebra)|basis]]. If (2,0,0) were replaced by (1,0,0), it would also form the [[standard basis|canonical basis]] of '''R'''<sup>3</sup>.
 
Another spanning set for the same space is given by {(1,2,3), (0,1,2), (&minus;1,1/2,3), (1,1,1)}, but this set is not a basis, because it is [[Linear dependency|linearly dependent]].
 
The set {(1,0,0), (0,1,0), (1,1,0)} is not a spanning set of '''R'''<sup>3</sup>; instead its span is the space of all vectors in '''R'''<sup>3</sup> whose last component is zero.
 
==Theorems==
 
'''Theorem 1:''' The subspace spanned by a non-empty subset ''S'' of a vector space ''V'' is the set of all linear combinations of vectors in ''S''.
 
This theorem is so well known that at times it is referred to as the definition of span of a set.  
 
'''Theorem 2:''' Every spanning set ''S'' of a vector space ''V'' must contain at least as many elements as any [[Linear independence|linearly independent]] set of vectors from ''V''.
 
'''Theorem 3:''' Let ''V'' be a finite dimensional vector space. Any set of vectors that spans ''V'' can be reduced to a basis for ''V'' by discarding vectors if necessary (i.e. if there are linearly dependent vectors in the set). If the [[axiom of choice]] holds, this is true without the assumption that ''V'' has finite dimension.
 
This also indicates that a basis is a minimal spanning set when ''V'' is finite dimensional.
 
==Closed linear span==
In [[functional analysis]], a closed linear span of a [[Set (mathematics)|set]] of [[vector space|vectors]] is the minimal closed set which contains the linear span of that set.
Suppose that ''X'' is a normed vector space and let ''E'' be any non-empty subset of ''X''. The '''closed linear span''' of ''E'', denoted by <math>\overline{\operatorname{Sp}}(E)</math> or <math>\overline{\operatorname{Span}}(E)</math>, is the intersection of all the closed linear subspaces of ''X'' which contain ''E''.
 
One mathematical formulation of this is
 
:<math>\overline{\operatorname{Sp}}(E)=\{u\in X | \forall\epsilon>0\,\exists x\in\operatorname{Sp}(E) : \|x-u\|<\epsilon\}.</math>
 
==Notes==
 
The linear span of a set is dense in the closed linear span. Moreover, as stated in the below lemma, the closed linear span is indeed the [[closure (mathematics)|closure]] of the linear span.
 
Closed linear spans are important when dealing with closed linear subspaces (which are themselves highly important, consider [[Riesz's lemma]]).
 
==A useful lemma==
 
Let ''X'' be a normed space and let ''E'' be any non-empty subset of ''X''. Then
 
(a) <math>\overline{\operatorname{Sp}}(E)</math> is a closed linear subspace of ''X'' which contains ''E'',
 
(b) <math>\overline{\operatorname{Sp}}(E)=\overline{\operatorname{Sp}(E)}</math>, viz. <math>\overline{\operatorname{Sp}}(E)</math> is the closure of <math>\operatorname{Sp}(E)</math>,
 
(c) <math>E^\perp=(\operatorname{Sp}(E))^\perp=(\overline{\operatorname{Sp}(E)})^\perp.</math>
 
(So the usual way to find the closed linear span is to find the linear span first, and then the closure of that linear span.)
 
==Matroids==
 
Generalizing the definition of the span of points in space, a subset ''X'' of the ground set of a [[matroid]] is called a ''spanning set'' if the rank of ''X'' equals the rank of the entire ground set.
 
==See also==
*[[Affine hull]]
*[[Convex hull]]
 
==External links==
 
* [https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/linear_combinations/v/linear-combinations-and-span Linear Combinations and Span: Understanding linear combinations and spans of vectors], khanacademy.org.
 
==References==
* {{springer|author=M.I. Voitsekhovskii|title=Linear hull|id=L/l059260}}
* {{cite web |url=http://www.math.ucdavis.edu/~anne/mat67_course_notes.pdf |title=Linear Algebra - As an Introduction to Abstract Mathematics||last1=Lankham |first1=Isaiah |last2=Nachtergaele |first2=Bruno |last2=Schilling |first2=Anne |publisher=University of California, Davis |date= 13 February 2010 |accessdate=27 September 2011 }}
* Rynne & Youngson (2001). ''Linear functional analysis'', Springer.
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[[Category:Abstract algebra]]
[[Category:Linear algebra]]
 
{{Linear algebra}}
 
[[pl:Podprzestrzeń liniowa#Powłoka liniowa]]
[[ru:Векторное пространство#Линейная оболочка]]

Latest revision as of 12:06, 26 December 2014

It's a rough road to travel and most adults have traveled it at least once. Few of us practice patience and for some of us it comes naturally. The law of attraction shows us that what happens to us are usually the things that we let our minds think. These fragrant-free, activated carbon sheets come in a set of five and absorb unnecessary scents and humidity from seeping into clothing. He may have another lover in his life right now, but if he showed signs of interest in you, it means he enjoyed the relationship he once had with you.

You want to improve yourself, improve your life and get back the love you thought you lost. Your ex is probably in a hurry to get over you and is not clear on what he or she feels. Perhaps it was at a party where you were both flirting from across the room. During your time apart analyze what the problems were and if you need to, get some counseling to help you figure out what went wrong. From that point, I suggest that you look through the upper area for a few things to loot and an ebook.

In the meantime, there are things you can do when your ex has a new girlfriend. Sure you're not officially "dating" anymore, but eventually you'll get her back. It takes a long time to establish a deep connection so it only makes sense that it would take a long time to untangle the connection. Be careful not to analyze things too much, because over analyzing may prevent you from acting the right way when trying to figure out "what can I do to get my ex girlfriend back". Detaching yourself from him gives you an air of self-reliance and by not forcing the issue you make it appear as if you've accepted the separation.

You can become a stronger and more attractive woman through it all. As far as possible, cut off all possible lines of contact or communication with your ex. During the phone call, make sure you use this one simple trick that will make your ex unable to get you out of his mind. If he desires to see you again, agree but don't sound overzealous. Once you're on the rooftop, talk to the officers outside and then walk in.

If your ex had enough space they will eventually come around, realize that they miss you, and need to get back together. A thing funny, nevertheless romantic will work greatest in order to break the ice. While you shouldn't seem desperate, you have to let him know or perhaps the girl that you are attempting to try and fix the errors of the past. For any woman out there who is either interested in winning her guy back or perhaps scared of him breaking up with her before long, you can now lay your anxieties aside as Michael Webb's Getting Him Back provides you with the very best of advice that is guaranteed to assist you make him really yours. One of the most obvious methods of showing that you want to win back your boy or girl is by constantly telling him or her that you are still single.

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