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| [[File:SL(2,3); Cayley table.svg|thumb|[[Cayley table]] of SL(2,3).]]
| | For over a decade, the Prostate Cancer Education Council (PCEC) has provided free and low-cost screenings to guys during the third week of September, called Prostate Cancer Awareness Week (PCAW).<br><br>Number 6. Visit the venue before the marriage date, when possible. May it be from the easy yard of the groom or the cliff of an island, it's highly suggested to visit the destination where the ceremony usually be held. This usually aid you tremendously for it we will understand what to anticipate plus tackle. Simply observe the step, assuming you'll shoot at the mountains.<br><br>If we had a magic fat loss machine -- which you stepped inside and pushed a button and you came out lookin waist to height ratio EXACTLY how you've always dreamed of looking and feeling - the exact dress or pant size we wanted, with the definition plus muscle tone we desire, at the bodyfat percentage you need - you are able to see the abs (if that's what you want) etc etc.Would we be interested? Of course you'd!!But what if the side effect of the fat loss machine is the fact that it increased bone density and muscle density by 100%. So while you looked plus felt better than you've ever felt before -- the scale is 50lbs high than it's ever been. So for example a 150lb-er would come out hunting amazing, however weighing 200lbs...<br><br>Number 2. Risk. Try shots and themes which you have not done before or are not comfortable with. Many seasoned marriage photographers recommend this technique not only for weddings. Developing the distinctive style is not a one-time thing, keep on trying new different designs and youll be surprise how extraordinary your pictures might look like.<br><br>The ideal technique to work diesel tuning on present day vehicles is to personalize the software in the engine control unit, or ECU. The ECU handles every element of engine procedure, control data from various sensors plus informing the fuel injection system how to proceed. The ECU's software programmes, or maps, control fuel delivery and timing so that the engine runs effectively at any RPM and pollution levels remain within satisfactory levels. The first generation of engine control units kept their maps on a removable computer chip. Newer ECUs have built-in maps which could be accessed through the On Board Diagnostic port.<br><br>According to the article "What Would the Real Barbie Look Like" in the BBC News Magazine on Friday March 2, 2009 when Barbie were 5'6" then her waist would be 20", her hips 29" plus bust a mere 27". The nearest real people to people measurements are Victoria Beckham with a 23" waist and Brigitte Bardot with a 20" waist. Both women are really well-known, quite skinny plus not your average. According to the measurements of [http://safedietplansforwomen.com/waist-to-height-ratio waist to height ratio] a Barbie figure will be achieved by 1 inside 100,000 ladies and a Ken figure might be accomplished in one inside fifty men.<br><br>Those whom were overweight were 22% more probably to have an ischemic stroke than general weight people; overweight topics were 64% more probably to have a stroke. The hemorrhagic stroke risk wasn't any high for overweight topics nevertheless was 24% higher for those who were fat.<br><br>Using weight resistance exercises to build muscle and excellent intensity interval cardio training to lower body fat is not anything modern and all qualified personal trainers ought to be capable to plan a superior system for we. Plus contemporary advances in nutrition, you need to be able to your desired bodyshape sooner instead of later. So plan your workouts plus nutrition with a well-defined objective of getting a swimmer's physique plus not which of the bodybuilder and we will soon get the admirable guy pro swimmer's build. |
| {{Group theory sidebar |Topological}}
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| {{Lie groups |Classical}}
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| In [[mathematics]], the '''special linear group''' of degree ''n'' over a [[Field (mathematics)|field]] ''F'' is the set of {{nowrap|''n'' × ''n''}} [[Matrix (mathematics)|matrices]] with [[determinant]] 1, with the group operations of ordinary [[matrix multiplication]] and [[matrix inversion]]. This is the [[normal subgroup]] of the [[general linear group]] given by the [[kernel (algebra)|kernel]] of the determinant
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| :<math>\det\colon \operatorname{GL}(n, F) \to F^\times.</math>
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| where we write ''F''<sup>×</sup> for the multiplicative group of ''F'' (that is, excluding 0).
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| These elements are "special" in that they fall on a [[Algebraic variety|subvariety]] of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries).
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| ==Geometric interpretation==
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| The special linear group {{nowrap|SL(''n'', '''R''')}} can be characterized as the group of ''[[volume]] and [[orientation (mathematics)|orientation]] preserving'' linear transformations of '''R'''<sup>''n''</sup>; this corresponds to the interpretation of the determinant as measuring change in volume and orientation.
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| ==Lie subgroup==
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| When ''F'' is '''R''' or '''C''', {{nowrap|SL(''n'', ''F'')}} is a [[Lie subgroup]] of {{nowrap|GL(''n'', ''F'')}} of dimension {{nowrap|''n''<sup>2</sup> − 1}}. The [[Lie algebra]] <math>\mathfrak{sl}(n, F)</math> of SL(''n'', ''F'') consists of all {{nowrap|''n'' × ''n''}} matrices over ''F'' with vanishing [[trace (matrix)|trace]]. The Lie bracket is given by the [[commutator]].
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| ==Topology==
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| Any invertible matrix can be uniquely represented according to the [[polar decomposition]] as the product of a [[unitary matrix]] and a [[hermitian matrix]] with positive [[eigenvalue]]s. The [[determinant]] of the unitary matrix is on the [[unit circle]] while that of the hermitian matrix is real and positive and, since, in the case of a matrix from the special linear group the product of these two determinants must be 1, then each of them must be 1. Therefore, a special linear matrix can be written as the product of a [[special unitary matrix]] (or [[special orthogonal matrix]] in the real case) and a [[Positive-definite matrix|positive definite]] hermitian matrix (or [[symmetric matrix]] in the real case) having determinant 1.
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| Thus the topology of the group {{nowrap|SL(''n'', '''C''')}} is the [[product topology|product]] of the topology of SU(''n'') and the topology of the group of hermitian matrices of unit determinant with positive eigenvalues. A hermitian matrix of unit determinant and having positive eigenvalues can be uniquely expressed as the [[matrix exponential|exponential]] of a [[traceless]] hermitian matrix, and therefore the topology of this is that of {{nowrap|(''n''<sup>2</sup> − 1)}}-dimensional [[Euclidean space]].
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| The topology of {{nowrap|SL(''n'', '''R''')}} is the product of the topology of [[special orthogonal matrix|SO]](''n'') and the topology of the group of symmetric matrices with positive eigenvalues and unit determinant. Since the latter matrices can be uniquely expressed as the exponential of symmetric traceless matrices, then this latter topology is that of {{nowrap|(''n'' + 2)(''n'' − 1)/2}}-dimensional Euclidean space.
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| The group {{nowrap|SL(''n'', '''C''')}}, like SU(''n''), is simply connected while {{nowrap|SL(''n'', '''R''')}}, like SO(''n''), is not. {{nowrap|SL(''n'', '''R''')}} has the same fundamental group as {{nowrap|GL<sup>+</sup>(''n'', '''R''')}} or SO(''n''), that is, '''Z''' for {{nowrap|1=''n'' = 2}} and '''Z'''<sub>2</sub> for {{nowrap|''n'' > 2}}.
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| ==Relations to other subgroups of GL(''n'',''A'')==
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| {{see also|Whitehead's lemma}}
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| Two related subgroups, which in some cases coincide with SL, and in other cases are accidentally conflated with SL, are the [[commutator subgroup]] of GL, and the group generated by [[Shear mapping|transvections]]. These are both subgroups of SL (transvections have determinant 1, and det is a map to an abelian group, so [GL, GL] ≤ SL), but in general do not coincide with it.
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| The group generated by transvections is denoted {{nowrap|E(''n'', ''A'')}} (for [[elementary matrices]]) or {{nowrap|TV(''n'', ''A'')}}. By the second [[Steinberg relations|Steinberg relation]], for {{nowrap|''n'' ≥ 3}}, transvections are commutators, so for {{nowrap|''n'' ≥ 3}}, {{nowrap|E(''n'', ''A'') ≤ [GL(''n'', ''A''), GL(''n'', ''A'')]}}. | |
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| <!-- Does equality hold? Dunno. -->
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| For {{nowrap|1=''n'' = 2}}, transvections need not be commutators (of {{nowrap|2 × 2}} matrices), as seen for example when ''A'' is '''F'''<sub>2</sub>, the field of two elements, then
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| :<math>\operatorname{Alt}(3) \cong [\operatorname{GL}(2, \mathbf{F}_2),\operatorname{GL}(2, \mathbf{F}_2)] < \operatorname{E}(2, \mathbf{F}_2) = \operatorname{SL}(2, \mathbf{F}_2) = \operatorname{GL}(2, \mathbf{F}_2) \cong \operatorname{Sym}(3),</math>
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| where Alt(3) and Sym(3) denote the [[alternating group|alternating]] resp. [[symmetric group]]<!--- I suppose this is meant; that article does not mention "Sym(n)" notation---> on 3 letters.
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| However, if ''A'' is a field with more than 2 elements, then {{nowrap|1=E(2, ''A'') = [GL(2, ''A''), GL(2, ''A'')]}}, and if ''A'' is a field with more than 3 elements, {{nowrap|1=E(2, ''A'') = [SL(2, ''A''), SL(2, ''A'')]}}.
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| In some circumstances these coincide: the special linear group over a field or a [[Euclidean domain]] is generated by transvections, and the ''stable'' special linear group over a [[Dedekind domain]] is generated by transvections. For more general rings the stable difference is measured by the [[special Whitehead group]] {{nowrap|1=SK<sub>1</sub>(''A'') := SL(''A'')/E(''A'')}}, where SL(''A'') and E(''A'') are the [[direct limit of groups|stable group]]s of the special linear group and elementary matrices.
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| ==Generators and relations==
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| If working over a ring where SL is generated by [[Shear mapping|transvections]] (such as a [[Field (mathematics)|field]] or [[Euclidean domain]]), one can give a [[presentation of a group|presentation]] of SL using transvections with some relations. Transvections satisfy the [[Steinberg relations]], but these are not sufficient: the resulting group is the [[Steinberg group (K-theory)|Steinberg group]], which is not the special linear group, but rather the [[universal central extension]] of the commutator subgroup of GL.
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| A sufficient set of relations for {{nowrap|SL(''n'', '''Z''')}} for {{nowrap|''n'' ≥ 3}} is given by two of the Steinberg relations, plus a third relation {{harv|Conder|Robertson|Williams|1992|p=19}}.
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| Let {{nowrap|1=''T<sub>ij</sub>'' := ''e<sub>ij</sub>''}}(1) be the elementary matrix with 1's on the diagonal and in the ''ij'' position, and 0's elsewhere (and ''i'' ≠ ''j''). Then
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| :<math>\begin{align}
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| \left[ T_{ij},T_{jk} \right] &= T_{ik} && \mbox{for } i \neq k\\
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| \left[ T_{ij},T_{kl} \right] &= \mathbf{1} && \mbox{for } i \neq l, j \neq k\\
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| (T_{12}T_{21}^{-1}T_{12})^4 &= \mathbf{1}\\
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| \end{align}</math>
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| are a complete set of relations for SL(''n'', '''Z'''), ''n'' ≥ 3.
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| ==Structure of GL(''n'',''F'')==
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| The group {{nowrap|GL(''n'', ''F'')}} splits over its determinant (we use {{nowrap|''F''<sup>×</sup> ≅ GL(1, ''F'') → GL(''n'', ''F'')}} as the [[monomorphism]] from ''F''<sup>×</sup> to {{nowrap|GL(''n'', ''F'')}}, see [[semidirect product]]), and therefore {{nowrap|GL(''n'', ''F'')}} can be written as a [[semidirect product]] of {{nowrap|SL(''n'', ''F'')}} by ''F''<sup>×</sup>: | |
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| :GL(''n'', ''F'') = SL(''n'', ''F'') ⋊ ''F''<sup>×</sup>.
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| ==See also==
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| * [[SL2(R)|SL(2, '''R''')]]
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| * [[SL2(C)|SL(2, '''C''')]]
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| * [[Modular group]]
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| * [[Projective linear group]]
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| ==References==
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| {{refimprove|date=January 2008}}
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| *{{Citation | last1=Conder | first1=Marston|author1-link=Marston Conder | last2=Robertson | first2=Edmund | last3=Williams | first3=Peter | title=Presentations for 3-dimensional special linear groups over integer rings | mr=1079696 | year=1992 | journal=Proceedings of the American Mathematical Society | volume=115 | issue=1 | pages=19–26 |doi=10.2307/2159559 | publisher=American Mathematical Society | jstor=2159559 }}
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| [[Category:Linear algebra]]
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| [[Category:Lie groups]]
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| [[pl:Pełna grupa liniowa#Specjalna grupa liniowa]]
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For over a decade, the Prostate Cancer Education Council (PCEC) has provided free and low-cost screenings to guys during the third week of September, called Prostate Cancer Awareness Week (PCAW).
Number 6. Visit the venue before the marriage date, when possible. May it be from the easy yard of the groom or the cliff of an island, it's highly suggested to visit the destination where the ceremony usually be held. This usually aid you tremendously for it we will understand what to anticipate plus tackle. Simply observe the step, assuming you'll shoot at the mountains.
If we had a magic fat loss machine -- which you stepped inside and pushed a button and you came out lookin waist to height ratio EXACTLY how you've always dreamed of looking and feeling - the exact dress or pant size we wanted, with the definition plus muscle tone we desire, at the bodyfat percentage you need - you are able to see the abs (if that's what you want) etc etc.Would we be interested? Of course you'd!!But what if the side effect of the fat loss machine is the fact that it increased bone density and muscle density by 100%. So while you looked plus felt better than you've ever felt before -- the scale is 50lbs high than it's ever been. So for example a 150lb-er would come out hunting amazing, however weighing 200lbs...
Number 2. Risk. Try shots and themes which you have not done before or are not comfortable with. Many seasoned marriage photographers recommend this technique not only for weddings. Developing the distinctive style is not a one-time thing, keep on trying new different designs and youll be surprise how extraordinary your pictures might look like.
The ideal technique to work diesel tuning on present day vehicles is to personalize the software in the engine control unit, or ECU. The ECU handles every element of engine procedure, control data from various sensors plus informing the fuel injection system how to proceed. The ECU's software programmes, or maps, control fuel delivery and timing so that the engine runs effectively at any RPM and pollution levels remain within satisfactory levels. The first generation of engine control units kept their maps on a removable computer chip. Newer ECUs have built-in maps which could be accessed through the On Board Diagnostic port.
According to the article "What Would the Real Barbie Look Like" in the BBC News Magazine on Friday March 2, 2009 when Barbie were 5'6" then her waist would be 20", her hips 29" plus bust a mere 27". The nearest real people to people measurements are Victoria Beckham with a 23" waist and Brigitte Bardot with a 20" waist. Both women are really well-known, quite skinny plus not your average. According to the measurements of waist to height ratio a Barbie figure will be achieved by 1 inside 100,000 ladies and a Ken figure might be accomplished in one inside fifty men.
Those whom were overweight were 22% more probably to have an ischemic stroke than general weight people; overweight topics were 64% more probably to have a stroke. The hemorrhagic stroke risk wasn't any high for overweight topics nevertheless was 24% higher for those who were fat.
Using weight resistance exercises to build muscle and excellent intensity interval cardio training to lower body fat is not anything modern and all qualified personal trainers ought to be capable to plan a superior system for we. Plus contemporary advances in nutrition, you need to be able to your desired bodyshape sooner instead of later. So plan your workouts plus nutrition with a well-defined objective of getting a swimmer's physique plus not which of the bodybuilder and we will soon get the admirable guy pro swimmer's build.