Ma trận (toán học) 123456

Ma trn (toỏn hc) MC LC i Mc lc nh ngha 1.1 ln L s Ký hiu Cỏc phộp toỏn c bn 4.1 Phộp cng, nhõn mt s vi ma trn, v ma trn chuyn v 4.2 Nhõn ma trn 4.3 Phộp toỏn hng 4.4 Ma trn Phng trỡnh tuyn tớnh Bin i tuyn tớnh 7 Ma trn vuụng 7.1 Cỏc loi thng gp 7.1.1 Ma trn chộo v ma trn tam giỏc 7.1.2 Ma trn n v 7.1.3 Ma trn i xng hoc i xng lch 7.1.4 Ma trn kh nghch v nghch o ca nú 7.1.5 Ma trn xỏc nh 7.1.6 Ma trn trc giao 10 Cỏc tớnh toỏn ch yu 10 7.2.1 Vt 10 7.2.2 nh thc 10 7.2.3 Ma trn nghch o 12 7.2.4 Vect riờng v tr riờng 12 7.2 Khớa cnh tớnh toỏn 12 Phõn tớ ma trn 13 10 Khớa cnh i s tru tng v tng quỏt húa 14 10.1 Ma trn vi cỏc phn t m rng 14 10.2 Mi liờn h vi ỏnh x tuyn tớnh 15 10.3 Nhúm ma trn 15 10.4 Ma trn rng 16 11 ng dng 16 11.1 Lý thuyt th 16 11.2 Gii tớch v hỡnh hc 17 ii MC LC 11.3 Lý thuyt xỏc sut v thng kờ 18 11.4 i xng v cỏc bin i vt lý hc 19 11.5 T hp tuyn tớnh ca cỏc trng thỏi lng t 19 11.6 Dao ng riờng 20 11.7 ang hỡnh hc 20 11.8 in t hc 20 12 am kho 20 13 am kho 24 13.1 am kho v vt lý 26 13.2 am kho v lch s 26 14 Liờn kt ngoi 26 15 Ngun, ngi úng gúp, v giy phộp o bn v hỡnh nh 28 15.1 Vn bn 28 15.2 Hỡnh nh 28 15.3 Giy phộp ni dung 28 MC LC m-by-n matrix a2,1 a2,2 a2,3 a3,1 a3,2 a3,3 a1,3 a1,2 i c h a n g e s a1,1 m rows j changes n columns ai,j Mi phn t ca mt ma trn thng c ký hiu bng mt bin vi hai ch s di Vớ d, a2,1 biu din phn t hng th hai v ct th nht ca ma trn A Trong toỏn hc, ma trn l mt mng ch nht[1] cỏc s, ký hiu, hoc biu thc, sp xp theo hng v ct [2][3] m mi ma trn tuõn theo nhng quy tc nh trc Tng ụ ma trn c gi l cỏc phn t hoc mc Vớ d mt ma trn cú hng v ct [ 20 ] 13 Khi cỏc ma trn cú cựng kớch thc (chỳng cú cựng s hng v cựng s ct), thỡ cú th thc hin phộp cng hoc tr hai ma trn trờn cỏc phn t tng ng ca chỳng Tuy vy, quy tc ỏp dng cho phộp nhõn ma trn ch cú th thc hin c ma trn th nht cú s ct bng s hng ca ma trn th hai ng dng chớnh ca ma trn ú l phộp biu din cỏc bin i tuyn tớnh, tc l s tng quỏt húa hm tuyn tớnh nh f (x) = 4x Vớ d, phộp quay cỏc vect khụng gian ba chiu l mt phộp bin i tuyn tớnh m cú th biu din bng mt ma trn quay R: nu v l vect ct (ma trn ch cú mt ct) miờu t v trớ ca mt im khụng gian, tớch ca Rv l mt vec t ct miờu t v trớ ca im ú sau phộp quay ny Tớch ca hai ma trn bin i l mt ma trn biu din hp ca hai phộp bin i tuyn tớnh Mt ng dng khỏc ca ma trn ú l tỡm nghim ca cỏc h phng trỡnh tuyn tớnh Nu l ma trn vuụng, cú th thu c mt s tớnh cht ca nú bng cỏch tớnh nh thc ca nú Vớ d, ma trn vuụng l ma trn kh nghch nu v ch nu nh thc ca nú khỏc khụng an nim hỡnh hc ca mt phộp bin i tuyn tớnh l nhn c (cựng vi nhng thụng tin khỏc) t tr riờng v vec t riờng ca ma trn Cú th thy ng dng ca lý thuyt ma trn hu ht cỏc lnh vc khoa hc Trong mi nhỏnh ca vt lý hc, bao gm c hc c in, quang hc, in t hc, c hc lng t, v in ng lc hc lng t, chỳng c s dng nghiờn cu cỏc hin tng vt lý, nh chuyn ng ca vt rn Trong mỏy tớnh, ma trn c s dng chiu mt nh chiu lờn mn hỡnh chiu Trong lý thuyt xỏc sut v thng kờ, cỏc ma trn ngu 2 LCH S nhiờn c s dng miờu t hp cỏc xỏc sut; vớ d, chỳng dựng thut toỏn PageRank xp hng cỏc trang lnh tỡm kim ca Google.[4] Phộp tớnh ma trn tng quỏt húa cỏc khỏi nim gii tớch nh o hm v hm m i vi s chiu ln hn Mt nhỏnh chớnh ca gii tớch s dnh phỏt trin cỏc thut toỏn hu hiu cho cỏc tớnh toỏn ma trn, mt ch ó hng trm nm tui v l mt lnh vc nghiờn cu rng ngy Phng phỏp khai trin ma trn lm n gin húa cỏc tớnh toỏn c v mt lý thuyt ln thc hnh Nhng thut toỏn da trờn nhng cu trỳc ca cỏc ma trn c bit, nh ma trn tha (sparse) v ma trn gn chộo, giỳp gii quyt nhng tớnh toỏn phng phỏp phn t hu hn v nhng tớnh toỏn khỏc Ma trn vụ hn xut hin c hc thiờn th v lý thuyt nguyờn t Mt vớ d n gin v ma trn vụ hn l ma trn biu din cỏc toỏn t o hm, m tỏc dng n chui Taylor ca mt hm s nh ngha Ma trn l mt mng ch nht cha cỏc s hoc nhng i tng toỏn hc khỏc, m cú th nh ngha mt s phộp toỏn nh cng hoc nhõn trờn cỏc ma trn.[5] Hay gp nht ú l ma trn trờn mt trng F l mt mng ch nht cha cỏc i lng vụ hng ca F.[6][7] Bi vit ny cp n cỏc ma trn thc v phc, tc l cỏc ma trn m cỏc phn t ca nú l nhng s thc hoc s phc Nhng loi ma trn tng quỏt hn c tho lun bờn di Vớ d, ma trn thc: 1, 0, 5, A = 20, 9, 6, Cỏc s, ký hiu hay biu thc ma trn c gi l cỏc phn t ca nú Cỏc ng theo phng ngang hoc phng dc cha cỏc phn t ma trn c gi tng ng l hng v ct 1.1 ln ln hay c ca ma trn c nh ngha bng s lng hng v ct m ma trn cú Mt ma trn m hng v n ct c gi l ma trn m ì n hoc ma trn m-nhõn-n, m v n c gi l chiu ca nú Vớ d, ma trn A trờn l ma trn ì Ma trn ch cú mt hng gi l vect hng, v nhng ma trn ch cú mt ct gi l vect ct Ma trn cú cựng s hng v s ct c gi l ma trn vuụng Ma trn cú vụ hn s hng hoc s ct (hoc c hai) c gi l ma trn vụ hn Trong mt s trng hp, nh chng trỡnh i s mỏy tớnh, s cú ớch xột mt ma trn m khụng cú hng hoc khụng cú ct, goi l ma trn rng Lch s Ma trn cú mt lch s di v ng dng gii cỏc phng trỡnh tuyn tớnh nhng chỳng c bit n l cỏc mng cho ti tn nhng nm 1800 Cun sỏch Cu chng toỏn thut vit vo khong nm 152 TCN a phng trn gii h nm phng trỡnh tuyn tớnh,[8] bao gm khỏi nim v nh thc Nm 1545 nh toỏn hc ngi í Girolamo Cardano gii thiu phng phỏp gii ny vo chõu u ụng cụng b quyn Ars Magna.[9] Nh toỏn hc Nht Bn Seki ó s dng phng phỏp mng ny gii h phng trỡnh vo nm 1683.[10] Nh toỏn hc H Lan Jan de Wi ln u tiờn biu din cỏc bin i di dng ma trn mng cun sỏch vit nm 1659 Elements of Curves (1659).[11] Gia cỏc nm 1700 v 1710 Gofried Wilhelm Leibniz cụng b phng phỏp s dng cỏc mng ghi li thụng tin hay tỡm nghim v nghiờn cu trờn 50 loi ma trn khỏc nhau.[9] Cramer a quy tc ca ụng vo nm 1750 ut ng ting Anh matrix (ting Latin l womb, dn xut t materm[12] ) James Joseph Sylvester nờu vo nm 1850,[13] ụng nhn rng ma trn l mt i tng lm xut hin mt s nh thc m ngy gi l phn ph i s, tc l nh thc ca nhng ma trn nh hn thu c t ma trn ban u bng cỏch xúa i cỏc hng v cỏc ct Trong mt bi bỏo nm 1851, Sylvester gii thớch: Tụi ó nh ngha bi bỏo trc v Ma trn l mt mng ch nht cha cỏc phn t, m nhng nh thc khỏc cú th a nh thc ca ma trn m.[14] Arthur Cayley ng mt chuyờn lun v cỏc phộp bin i hỡnh hc s dng ma trn ngoi nhng phộp bin i quay ó c kho sỏt trc ú ay vo ú, ụng nh ngha cỏc phộp toỏn nh cng, tr, nhõn v chia nhng ma trn ny v chng t cỏc quy tc kt hp v phõn phi c tha Cayley ó nghiờn cu v minh chng tớnh cht khụng giao hoỏn ca phộp nhõn ma trn cng nh tớnh giao hoỏn ca phộp cng ma trn.[9] Lý thuyt ma trn s khai b gii hn cỏch s dng cỏc mng v tớnh nh thc v cỏc phộp toỏn ma trn tru tng ca Arthur Cayley ó lm nờn cuc cỏch mng cho lý thuyt ny ễng ỏp dng khỏi nim ma trn cho h phng trỡnh tuyn tớnh c lp Nm 1858 Cayley cụng b Hi ký v lý thuyt ma trn[15][16] ú ụng nờu v chng minh nh lý Cayley-Hamilton.[9] Nh toỏn hc ngi Anh Cullis l ngi u tiờn s dng ký hiu ngoc hin i cho ma trn vo nm 1913 v ụng cng vit ký hiu quan trng A = [a,] biu din mt ma trn vi a, l phn t hng th i v ct th j.[9] ỏ trỡnh nghiờn cu nh thc xut phỏt t mt s ngun khỏc nhau.[17] Cỏc bi toỏn s hc dn Gauss i ti liờn h cỏc h s ca dng ton phng, nhng a thc cú dng x + xy 2y , v ỏnh x tuyn tớnh khụng gian ba chiu vi ma trn Eisenstein ó phỏt trin xa hn cỏc khỏi nim ny, vi nhn xột theo cỏch phỏt biu hin i rng tớch ma trn l khụng giao hoỏn Cauchy l ngi u tiờn chng minh nhng mnh tng quỏt v nh thc, ụng s dng nh ngha nh sau v nh thc ca ma trn A = [a,]: thay th ly tha ak bng a a thc a1 a2 ã ã ã an (aj ) i m, v nu hng ca ma trn Jacobi t giỏ tr ln nht bng m, f l hm kh nghch ti im ú theo nh nh lý hm n.[94] 18 11 Ti im yờn nga (x = 0, y = 0) () ca hm f(x,y) = x2 y2 , ma trn Hess [ NG DNG ] l khụng xỏc nh Cỏc nh toỏn hc cú th phõn loi phng trỡnh o hm riờng bng cỏch xột ma trn cỏc h s ca nhng toỏn t vi phõn bc cao nht ca phng trỡnh i vi phng trỡnh o hm riờng eliptic ma trn ny xỏc nh dng v cú nh hng quyt nh n hp nghim kh d ca bi toỏn tỡm nghim phng trỡnh o hm riờng.[95] Phng phỏp phn t hu hn l mt phng phỏp s quan trng gii phng trỡnh o hm riờng, c ng dng rng rói vic mụ phng cỏc h thng thc phc hp Phng phỏp ny ỏnh giỏ xp x nghim ca phng trỡnh bng cỏch phõn chia phng trỡnh thnh cỏc hm tuyn tớnh, m nhng hm ny c chn li to mn, m t ú cú th vit phng trỡnh di dng phng trỡnh ma trn.[96] 11.3 Lý thuyt xỏc sut v thng kờ Ma trn quỏ trỡnh ngu nhiờn l nhng ma trn vuụng m cỏc hng ca nú l cỏc vect xỏc sut, tc l vect cú cỏc thnh phn khụng õm v tng ca chỳng bng Ma trn ngu nhiờn c s dng tỡm xớch Markov vi nhng trng thỏi hu hn.[97] Mt hng ca ma trn ngu nhiờn cho phõn b xỏc sut ca v trớ tip theo ca mt s ht trng thỏi tng ng vi hng ú Cỏc tớnh cht ca xớch Markov ging nh im hp dn (aractor), nhng im trng thỏi m cỏc ht cui cựng t ti, cú th c suy t nhng vect riờng ca ma trn chuyn tip.[98] Lý thuyt thng kờ cng ỏp dng ma trn nhiu dng khỏc nhau.[99] ng kờ mụ t cp ti hp d liu c mụ t, m chỳng c biu din bng cỏc ma trn d liu, sau ú cỏc nh thng kờ s dng nhng k thut thu gim s bin (dimensionality reduction kho sỏt cỏc ma trn ny Ma trn hip phng sai mó húa phng sai tng h ca cỏc bin ngu nhiờn.[100] Cỏc k thut khỏc s dng ma trn l bỡnh phng ti thiu, mt phng phỏp xp x hp hu hn nhng cp im (x , y ), (x , y ),, (xN , yN ), bng mt hm s tuyn tớnh y ax + b, i = 1,, N chỳng cú th c thit lp da trờn ngụn ng ca lý thuyt ma trn, 11.4 i xng v cỏc bin i vt lý hc 19 Hai xớch Markov khỏc [th th hin ] s ht[(trong tng ] s 1000) trng thỏi C hai giỏ tr gii hn c xỏc nh t 7 ma trn chuyn tip, ln lt l () v (en) .3 vi liờn h n k thut phõn tớch thnh tớch cỏc ma trn giỏ tr riờng c bit (singular value decomposition).[101] Ma trn ngu nhiờn l ma trn vi phn t l nhng s ngu nhiờn, phự hp cho nghiờn cu tớnh cht phõn b xỏc sut, nh l ma trn phõn b chun Ngoi lý thuyt xỏc sut, chỳng cũn c ỏp dng phm vi t lý thuyt s ti vt lý hc.[102][103] 11.4 i xng v cỏc bin i vt lý hc Cỏc bin i tuyn tớnh v nhng i xng i kốm úng vai trũ quan trng vt lý hin i Vớ d, cỏc ht c bn lý thuyt trng lng t c phõn loi nh nhng biu din ca nhúm Lorentz thuyt tng i hp v, c th hn, bi ng x ca chỳng di nhúm spin Nhng biu din tng minh bao gm ma trn Pauli v ma trn gamma tng quỏt hn l phn tớch phõn ca miờu t vt lý i vi fermion, m hot ng nh l spinor.[104] i vi ba loi quark nh nht, cú th biu din chỳng bng nhúm unita c bit SU(3); v cỏc nh vt lý s dng ma trn biu din thun tin gi l ma trn Gell-Mann tớnh toỏn liờn quan, ma trn ny cng c s dng cho nhúm chun SU(3) m nú tr thnh c s cho lý thuyt miờu t v tng tỏc mnh, sc ng lc hc lng t Ma trn CabibboKobayashiMaskawa, biu din trng thỏi c bn cỏc quark tham gia vo tng tỏc yu, nú khụng ging nh ma trn Gell-Mann, nhng cú liờn h tuyn tớnh vi trng thỏi c bn cỏc quark xỏc nh lờn ht t hp vi tớnh cht v lng c th.[105] 11.5 T hp tuyn tớnh ca cỏc trng thỏi lng t Mụ hỡnh u tiờn v c hc lng t (Heisenberg, 1925) biu din cỏc toỏn t ca lý thuyt bng cỏc ma trn vụ hn chiu tỏc dng lờn cỏc trng thỏi lng t.[106] Lý thuyt ny cũn c gi l c hc ma trn Mt vớ d c 20 12 THAM KHO th ú l ma trn mt c trng cho trng thỏi trn ca mt h lng t nh l t hp tuyn tớnh ca cỏc trng thỏi riờng thun tuý v c bn.[107] Vớ d khỏc v ma trn tr thnh cụng c quan trng cho miờu t cỏc thớ nghim tỏn x l hot ng trung tõm ca vt lý ht thc nghim: Nhng phn ng va chm xy cỏc mỏy gia tc, ni cỏc ht c cho va chm i u vo mt va chm nh, vi kt qu sau va chm sinh nhng ht mi, cú th c miờu t bng tớch vụ hng ca trng thỏi nhng ht hỡnh thnh vi t hp tuyn tớnh ca cỏc ht tham gia vo va chm T hp tuyn tớnh ny cho bi ma trn gi l ma trn S, nú cha mi thụng tin v cỏc tng tỏc kh d gia nhng ht tham gia vo va chm.[108] 11.6 Dao ng riờng ng dng ph bin ca ma trn vt lý hc l dựng miờu t h dao ng iu hũa tuyn tớnh Phng trỡnh chuyn ng ca nhng h ny cú th miờu t theo dng ma trn, vi ma trn lng nhõn vi mt vect ta s cho s hng ng hc, ma trn lc nhõn vi vect chuyn di v trớ s cho c trng ca tng tỏc Cỏch tt nht thu c nghim ca h phng trỡnh ú l xỏc nh cỏc vect riờng ca h, hay cỏc dao ng riờng, bng cỏch chộo húa phng trỡnh ma trn Cỏc k thut nh th ny l quan trng nghiờn nghiờn cu ni ng lc phõn t: cỏc dao ng bờn ca h cha cỏc nguyờn t thnh phn liờn kt vi nhau.[109] Chỳng cng cn thit miờu t dao ng c hc, dao ng mch in.[110] 11.7 Quang hỡnh hc ang hỡnh hc s dng cỏc ng dng ca ma trn nhiu hn Trong lý thuyt xp x ny, bn cht súng ca ỏnh sỏng c b qua Mụ hỡnh kt qu ú tia sỏng tr thnh tia hỡnh hc Nu s lch ca tia sỏng bi cỏc quang c l nh, tỏc dng ca mt thu kớnh hoc dng c phn x lờn mt tia sỏng cú th c biu din bng tớch ca mt vect hai thnh phn vi ma trn 2x2 gi l ma trn chuyn tip tia (ray transfer matrix): cỏc thnh phn ca vec t l dc ca tia sỏng v khong cỏch ca nú ti quang trc, ma trn mó húa cỏc tớnh cht ca quang c c s cú hai kiu ma trn, ú ma trn khỳc x miờu t s khỳc x ti b mt thu kớnh, v ma trn tnh tin miờu t s tnh tin ca mt phng tham chiu ti mt phng khỳc x k cn, ni mt ma trn khỳc x khỏc c ỏp dng ang h, bao gm t hp cỏc thu kớnh v cỏc dng c phn x, c miờu t n gin bng ma trn t tớch cỏc ma trn thnh phn.[111] 11.8 in t hc Phng phỏp phõn tớch dũng in vũng (mesh analysis) truyn thng in t hc dn ti vic tỡm nghim ca mt h phng trỡnh tuyn tớnh m cú th miờu t bng ma trn Hot ng ca nhiu linh kin in t c miờu t bng ma trn Nu A l mt vec t chiu vi cỏc thnh phn ca nú l in ỏp vo v v dũng vo i1 , gi B l mt vec t chiu vi cỏc thnh phn ca nú l in ỏp v v dũng i2 ỡ hot ng ca linh kin in t c miờu t bng phng trỡnh B = H A, vi H l ma trn x cha mt phn t tr khỏng (h12 ), v mt phn t dn (admitance) (h21 ) v hai i lng khụng th nguyờn (h11 v h22 ) Vic tớnh toỏn mch in thu v vic nhõn cỏc ma trn 12 Tham kho [1] hoc tng ng l bng [2] Anton (1987, tr 23) [3] Beauregard & Fraleigh (1973, tr 56) [4] K Bryan and T Leise e $25,000,000,000 eigenvector: e linear algebra behind Google SIAM Review, 48(3):569581, 2006 [5] Lang 2002 [6] Fraleigh (1976, tr 209) [7] Nering (1970, tr 37) 21 [8] Shen, Crossley & Lun 1999 cited by Bretscher 2005, p [9] Discrete Mathematics 4th Ed Dossey, Oo, Spense, Vanden Eynden, Published by Addison Wesley, ngy 10 thỏng 10 nm 2001 ISBN 978-0321079121 | p.564-565 [10] Needham, Joseph; Wang Ling (1959) Science and Civilisation in China III Cambridge: Cambridge University Press tr 117 ISBN 9780521058018 [11] Discrete Mathematics 4th Ed Dossey, Oo, Spense, Vanden Eynden, Published by Addison Wesley, ngy 10 thỏng 10 nm 2001 ISBN 978-0321079121 | p.564 [12] MerriamWebster dictionary, MerriamWebster, truy cp ngy 20 thỏng nm 2009 [13] Mc dự nhiu ngun cho rng J J Sylvester a thut ng matrix vo nm 1848, nhng Sylvester khụng cụng b ti liu no vo nm 1848 (V dn chng cho Sylvester khụng cụng b gỡ vo nm 1848, xem: J J Sylvester v H F Baker, ed., e Collected Mathematical Papers of James Joseph Sylvester (Cambridge, England: Cambridge University Press, 1904), vol 1.) Nm u tiờn m ụng s dng matrix xut hin vo nm 1850: J J Sylvester (1850) Additions to the articles in the September number of this journal, On a new class of theorems, and on Pascals theorem, e London, Edinburgh and Dublin Philosophical Magazine and Journal of Science, 37: 363-370 From page 369: For this purpose we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns is will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants " [14] e Collected Mathematical Papers of James Joseph Sylvester: 18371853, Paper 37, p 247 [15] Phil.Trans 1858, vol.148, pp.17-37 Math Papers II 475-496 [16] Dieudonnộ, ed 1978, Vol 1, Ch III, p 96 [17] Knobloch 1994 [18] Hawkins 1975 [19] Kronecker 1897 [20] Weierstrass 1915, pp 271286 [21] Bụcher 2004 [22] Mehra & Rechenberg 1987 [23] Oualline 2003, Ch [24] How to organize, add and multiply matrices - Bill Shillito TED ED Truy cp ngy thỏng nm 2015 [25] Brown 1991, Denition I.2.1 (addition), Denition I.2.4 (scalar multiplication), and Denition I.2.33 (transpose) [26] Brown 1991, eorem I.2.6 [27] Brown 1991, Denition I.2.20 [28] Brown 1991, eorem I.2.24 [29] Horn & Johnson 1985, Ch and [30] Bronson (1970, tr 16) [31] Kreyszig (1972, tr 220) [32] Proer & Morrey (1970, tr 869) [33] Kreyszig (1972, tr 241,244) [34] Schneider, Hans; 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Bingham, Christopher, MacAnova, University of Minnesota, School of Statistics, truy cp ngy 10 thỏng 12 nm 2008, a freeware package for matrix algebra and statistics Online matrix calculator, truy cp ngy 14 thỏng 12 nm 2009 Operation with matrices in R (determinant, track, inverse, adjoint, transpose) 28 15 NGUN, NGI ểNG GểP, V GIY PHẫP CHO VN BN V HèNH NH 15 Ngun, ngi úng gúp, v giy phộp cho bn v hỡnh nh 15.1 Vn bn Ma trn (toỏn hc) Ngun: https://vi.wikipedia.org/wiki/Ma_tr%E1%BA%ADn_(to%C3%A1n_h%E1%BB%8Dc)?oldid=25950522 Ngi úng gúp: Mxn, Mekong Bluesman, Vng Ngõn H, Trung, Chobot, YurikBot, Casablanca1911, Newone, DHN-bot, Ctmt, Viethavvh, JAnDbot, ijs!bot, Hong Cm, Haonhien, VolkovBot, TXiKiBoT, AlleborgoBot, SieBot, TVT-bot, Loveless, DragonBot, Tiepdinhvan, Qbot, BodhisavaBot, Nallimbot, Luckas-bot, Eternal Dragon, ArthurBot, Xqbot, Almabot, TobeBot, D'ohBot, Earthandmoon, Tnt1984, Namnguyenvn, TuHan-Bot, EmausBot, SweetLoveFC, RedBot, WikitanvirBot, Ripchip Bot, Movses-bot, Cheers!-bot, MerlIwBot, TuanUt, Alphama, AlphamaBot, Addbot, OctraBot, Aokiji, Khatrungan, itxongkhoiAWB, Tuanminh01, TuanminhBot, ẫn bc AWB, Knight167, Xixaxixup v 40 ngi vụ danh 15.2 Hỡnh nh Tp_tin:Area_parallellogram_as_determinant.svg Ngun: https://upload.wikimedia.org/wikipedia/commons/a/ad/Area_parallellogram_ as_determinant.svg Giy phộp: Public domain Ngi úng gúp: Own work, created with Inkscape Ngh s u tiờn: Jitse Niesen Tp_tin:Commons-logo.svg Ngun: https://upload.wikimedia.org/wikipedia/commons/4/4a/Commons-logo.svg Giy phộp: Public domain Ngi úng gúp: is version created by Pumbaa, using a proper partial circle and SVG geometry features (Former versions used to be slightly warped.) 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