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| In [[mathematics]], the '''dimension''' of a [[vector space]] ''V'' is the [[cardinal number|cardinality]] (i.e. the number of vectors) of a [[basis (linear algebra)|basis]] of ''V''.<ref>{{cite book|author=Itzkov, Mikhail|title=Tensor Algebra and Tensor Analysis for Engineers: With Applications to Continuum Mechanics|publisher=Springer|year=2009|isbn=978-3-540-93906-1|page=4|url=http://books.google.com/books?id=8FVk_KRY7zwC&pg=PA4}}</ref><ref>It is sometimes called '''Hamel dimension''' or '''algebraic dimension''' to distinguish it from other types of [[dimension]].</ref>
| | There's nothing like getting a good massage after a long day of doing hard work. Massaging the back can help you relax and can also help free your mind. Do you find this appealing? Read on if you answered yes.<br><br>Implement the bear hug technique to eliminate shoulder tension in your body. Use your arms to make an x across your chest. Put each hand on the opposing shoulder and rub firmly. This is how a quick, self massage can be given whenever you need it in your day.<br><br>Massage in an environment without loud noises. It's hard to relax when there's a lot of outside noise. Without quiet, the massage won't be able to reach its full purpose. If you desire to move, do so to maximize your comfort level. When you take the time to choose your location carefully, you will reap greater rewards.<br><br>If you have arthritis, you know the pain it causes. Medication helps, but it may not always take away the pain from aching bones. If you feel like your medication is not doing enough, consider getting a massage. It increases circulation and awakens the muscles.<br><br>Massages once or twice per week are a great idea. Many people find that getting regular massages helps to improve their mood and overall health. Massage is very relaxing, so it is understandable that it reduces stress. Try to go to the massage parlor twice a week, or more.<br><br>Try giving yourself a massage. Just use your thumbs to massage your muscles. Start at the legs and arms, working your way from bottom to top. Doing this massage after you wake up can rejuvenate your body for the day ahead. This helps to lessen stress and it can even help you sleep better.<br><br>Arthritis is a painful condition. Sometimes medication will work, but this is a more natural way to go about things. A massage can provide various benefits for your arthritis. Massages help pain because they increase circulation and flexibility.<br><br>Shiatsu massages come from Japan. There are many similarities to acupuncture; however, rather than using needles, fingers are used. If you have any thoughts concerning where by and how to use japanese massage ([http://wiki.acceed.de/index.php/Excellent_Advice_About_Giving_And_Getting_Massages wiki.acceed.de]), you can get in touch with us at the webpage. The massage therapist will put pressure on the pressure points in your body, making your body relax instantly. Shiatsu massages are for boosting energy levels and well being.<br><br>When you feel that you want a massage, tell the masseuse where your issues are. You are paying for a massage to ease away your aches and pains. Do not expect your therapist to find your problem areas right away and relieve the pain if you do not communicate and explain the kind of pain you are experiencing.<br><br>A Swedish or deep tissue massage can be rewarding for a first time massage experience. Other therapies may not prove as beneficial. The choices mentioned above should help you with most of your aches as you go through the process of learning more about your needs.<br><br>Remain calm and quiet while massaging someone. No one likes a chatty masseuse when you are trying to unwind and relax while getting a massage. There should be no sound, except possibly some nature sounds or soft music. Try to keep things very silent otherwise.<br><br>You are fortunate to find this particular article and the facts contained in it. Keeping up with new data is important. Continue reading articles and blogs of massage experts in order to avoid getting misled. |
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| For every vector space there exists a basis (if one assumes the [[axiom of choice]]), and all bases of a vector space have equal cardinality (see [[dimension theorem for vector spaces]]); as a result the dimension of a vector space is uniquely defined. We say ''V'' is '''finite-dimensional''' if the dimension of ''V'' is [[wiktionary:finite|finite]].
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| The dimension of the vector space ''V'' over the [[field (mathematics)|field]] ''F'' can be written as dim<sub>''F''</sub>(''V'') or as [V : F], read "dimension of ''V'' over ''F''". When ''F'' can be inferred from context, often just dim(''V'') is written.
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| == Examples ==
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| The vector space '''R'''<sup>3</sup> has
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| :<math>\left \{ \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} , \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} , \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \right \}</math>
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| as a [[Basis (linear algebra)|basis]], and therefore we have dim<sub>'''R'''</sub>('''R'''<sup>3</sup>) = 3. More generally, dim<sub>'''R'''</sub>('''R'''<sup>''n''</sup>) = ''n'', and even more generally, dim<sub>''F''</sub>(''F''<sup>''n''</sup>) = ''n'' for any [[field (mathematics)|field]] ''F''.
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| The [[complex number]]s '''C''' are both a real and complex vector space; we have dim<sub>'''R'''</sub>('''C''') = 2 and dim<sub>'''C'''</sub>('''C''') = 1. So the dimension depends on the base field.
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| The only vector space with dimension 0 is {0}, the vector space consisting only of its zero element.
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| == Facts ==
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| If ''W'' is a [[linear subspace]] of ''V'', then dim(''W'') ≤ dim(''V''). | |
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| To show that two finite-dimensional vector spaces are equal, one often uses the following criterion: if ''V'' is a finite-dimensional vector space and ''W'' is a linear subspace of ''V'' with dim(''W'') = dim(''V''), then ''W'' = ''V''.
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| '''R'''<sup>''n''</sup> has the standard basis {'''e'''<sub>1</sub>, ..., '''e'''<sub>''n''</sub>}, where '''e'''<sub>''i''</sub> is the ''i''-th column of the corresponding [[identity matrix]]. Therefore '''R'''<sup>''n''</sup>
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| has dimension ''n''.
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| Any two vector spaces over ''F'' having the same dimension are [[isomorphic]]. Any [[bijective]] map between their bases can be uniquely extended to a bijective linear map between the vector spaces. If ''B'' is some set, a vector space with dimension |''B''| over ''F'' can be constructed as follows: take the set ''F''<sup>(''B'')</sup> of all functions ''f'' : ''B'' → ''F'' such that ''f''(''b'') = 0 for all but finitely many ''b'' in ''B''. These functions can be added and multiplied with elements of ''F'', and we obtain the desired ''F''-vector space.
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| An important result about dimensions is given by the [[rank–nullity theorem]] for [[linear map]]s.
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| If ''F''/''K'' is a [[field extension]], then ''F'' is in particular a vector space over ''K''. Furthermore, every ''F''-vector space ''V'' is also a ''K''-vector space. The dimensions are related by the formula
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| :dim<sub>''K''</sub>(''V'') = dim<sub>''K''</sub>(''F'') dim<sub>''F''</sub>(''V'').
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| In particular, every complex vector space of dimension ''n'' is a real vector space of dimension 2''n''.
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| Some simple formulae relate the dimension of a vector space with the [[cardinality]] of the base field and the cardinality of the space itself.
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| If ''V'' is a vector space over a field ''F'' then, denoting the dimension of ''V'' by dim''V'', we have: | |
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| :If dim ''V'' is finite, then |''V''| = |''F''|<sup>dim''V''</sup>.
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| :If dim ''V'' is infinite, then |''V''| = max(|''F''|, dim''V'').
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| == Generalizations ==
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| One can see a vector space as a particular case of a [[matroid]], and in the latter there is a well-defined notion of dimension. The [[length of a module]] and the [[rank of an abelian group]] both have several properties similar to the dimension of vector spaces.
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| The [[Krull dimension]] of a commutative [[ring (algebra)|ring]], named after [[Wolfgang Krull]] (1899–1971), is defined to be the maximal number of strict inclusions in an increasing chain of [[prime ideal]]s in the ring.
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| === Trace ===
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| {{see also|Trace (linear algebra)}}
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| The dimension of a vector space may alternatively be characterized as the [[Trace (linear algebra)|trace]] of the [[identity operator]]. For instance, <math>\operatorname{tr}\ \operatorname{id}_{\mathbf{R}^2} = \operatorname{tr} \left(\begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{smallmatrix}\right) = 1 + 1 = 2.</math> This appears to be a circular definition, but it allows useful generalizations. | |
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| Firstly, it allows one to define a notion of dimension when one has a trace but no natural sense of basis. For example, one may have an algebra ''A'' with maps <math>\eta\colon K \to A</math> (the inclusion of scalars, called the ''unit'') and a map <math>\epsilon \colon A \to K</math> (corresponding to trace, called the ''[[counit]]''). The composition <math>\epsilon\circ \eta \colon K \to K</math> is a scalar (being a linear operator on a 1-dimensional space) corresponds to "trace of identity", and gives a notion of dimension for an abstract algebra. In practice, in [[bialgebra]]s one requires that this map be the identity, which can be obtained by normalizing the counit by dividing by dimension (<math>\epsilon := \textstyle{\frac{1}{n}} \operatorname{tr}</math>), so in these cases the normalizing constant corresponds to dimension.
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| Alternatively, one may be able to take the trace of operators on an infinite-dimensional space; in this case a (finite) trace is defined, even though no (finite) dimension exists, and gives a notion of "dimension of the operator". These fall under the rubric of "[[trace class]] operators" on a [[Hilbert space]], or more generally [[nuclear operator]]s on a [[Banach space]].
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| A subtler generalization is to consider the trace of a ''family'' of operators as a kind of "twisted" dimension. This occurs significantly in [[representation theory]], where the [[Character (mathematics)|character]] of a representation is the trace of the representation, hence a scalar-valued function on a [[group (mathematics)|group]] <math>\chi\colon G \to K,</math> whose value on the identity <math>1 \in G</math> is the dimension of the representation, as a representation sends the identity in the group to the identity matrix: <math>\chi(1_G) = \operatorname{tr}\ I_V = \dim V.</math> One can view the other values <math>\chi(g)</math> of the character as "twisted" dimensions, and find analogs or generalizations of statements about dimensions to statements about characters or representations. A sophisticated example of this occurs in the theory of [[monstrous moonshine]]: the [[j-invariant|''j''-invariant]] is the [[graded dimension]] of an infinite-dimensional graded representation of the [[Monster group]], and replacing the dimension with the character gives the [[McKay–Thompson series]] for each element of the Monster group.<ref>{{Harv|Gannon|2006}}</ref>
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| == See also ==
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| *[[Basis (linear algebra)]]
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| *[[Topological dimension]], also called Lebesgue covering dimension
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| *[[Fractal dimension]]
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| *[[Krull dimension]]
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| *[[Matroid rank]]
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| *[[Rank (linear algebra)]]
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| == References ==
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| {{reflist}}
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| {{refbegin}}
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| *{{Citation | first = Terry | last = Gannon | title = Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics | year = 2006 | isbn = 0-521-83531-3}}
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| {{refend}}
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| ==External links==
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| * [http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-9-independence-basis-and-dimension/ MIT Linear Algebra Lecture on Independence, Basis, and Dimension by Gilbert Strang] at MIT OpenCourseWare
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| {{DEFAULTSORT:Dimension (Vector Space)}}
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| [[Category:Linear algebra]]
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| [[Category:Dimension]]
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| [[Category:Vectors]]
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There's nothing like getting a good massage after a long day of doing hard work. Massaging the back can help you relax and can also help free your mind. Do you find this appealing? Read on if you answered yes.
Implement the bear hug technique to eliminate shoulder tension in your body. Use your arms to make an x across your chest. Put each hand on the opposing shoulder and rub firmly. This is how a quick, self massage can be given whenever you need it in your day.
Massage in an environment without loud noises. It's hard to relax when there's a lot of outside noise. Without quiet, the massage won't be able to reach its full purpose. If you desire to move, do so to maximize your comfort level. When you take the time to choose your location carefully, you will reap greater rewards.
If you have arthritis, you know the pain it causes. Medication helps, but it may not always take away the pain from aching bones. If you feel like your medication is not doing enough, consider getting a massage. It increases circulation and awakens the muscles.
Massages once or twice per week are a great idea. Many people find that getting regular massages helps to improve their mood and overall health. Massage is very relaxing, so it is understandable that it reduces stress. Try to go to the massage parlor twice a week, or more.
Try giving yourself a massage. Just use your thumbs to massage your muscles. Start at the legs and arms, working your way from bottom to top. Doing this massage after you wake up can rejuvenate your body for the day ahead. This helps to lessen stress and it can even help you sleep better.
Arthritis is a painful condition. Sometimes medication will work, but this is a more natural way to go about things. A massage can provide various benefits for your arthritis. Massages help pain because they increase circulation and flexibility.
Shiatsu massages come from Japan. There are many similarities to acupuncture; however, rather than using needles, fingers are used. If you have any thoughts concerning where by and how to use japanese massage (wiki.acceed.de), you can get in touch with us at the webpage. The massage therapist will put pressure on the pressure points in your body, making your body relax instantly. Shiatsu massages are for boosting energy levels and well being.
When you feel that you want a massage, tell the masseuse where your issues are. You are paying for a massage to ease away your aches and pains. Do not expect your therapist to find your problem areas right away and relieve the pain if you do not communicate and explain the kind of pain you are experiencing.
A Swedish or deep tissue massage can be rewarding for a first time massage experience. Other therapies may not prove as beneficial. The choices mentioned above should help you with most of your aches as you go through the process of learning more about your needs.
Remain calm and quiet while massaging someone. No one likes a chatty masseuse when you are trying to unwind and relax while getting a massage. There should be no sound, except possibly some nature sounds or soft music. Try to keep things very silent otherwise.
You are fortunate to find this particular article and the facts contained in it. Keeping up with new data is important. Continue reading articles and blogs of massage experts in order to avoid getting misled.