A system of lwento zibalo Aljibhra. uhlelo eyohlobo of lwento zibalo algebraic

Esikoleni, ngamunye wethu wafunda ezothando futhi, impela, uhlelo of zibalo. Kodwa ababaningi abantu abazi ukuthi kukhona izindlela eziningana ukuzixazulula. Namuhla sizobona ncamashi zonke izindlela ukuxazulula uhlelo of zibalo lwento algebraic, okuyinto akhiwa zibalo engaphezu kwemibili.

indaba

Namuhla siyazi ukuthi ubuciko ukuxazulula zibalo kanye izinhlelo zabo zivela eBabiloni lasendulo. Nokho, ukulingana ngefomu yabo ajwayelekile wabonakala nathi emva nesenzakalo setshwayo "=", okuyinto eyenzeka ngo-1556 ngu-isiZulu sezibalo irekhodi. By the way, lolu phawu wakhethwa ngesizathu: kusho izingxenye ezimbili okuhambisana alinganayo. Ngempela, isibonelo esihle kunazo zonke ukulingana akusho zibheda.

Umsunguli lettering yesimanje futhi ziwuphawu ngokwezinga ezingaziwa, sezibalo French Fransua Viet. Nokho, ukuqokwa sihluke kakhulu namuhla. Ngokwesibonelo, skwele kunombolo engaziwa ukuthi ebekwe Q incwadi (lat "quadratus".), Futhi cube - (. Lat "cubus") incwadi C. Letimphawu manje kubonakala kahle, kodwa kube indlela enembile kakhulu ukubhala simiso komugqa zibalo Aljibhra.

Nokho, kusimo izindlela owawukhona neqhinga ukuthi zezibalo kuxoxa ngazo kuphela izimpande omuhle. Mhlawumbe lokhu kungenxa yokuthi amanani negative Awunawo ngokoqobo. Ngandlela-thile, kodwa owokuqala kubhekwe izimpande ezimbi yathoma khona ngemva wezibalo Italian Niccolò Tartaglia, Gerolamo Cardano futhi Raphael Bombelli ngekhulu le-16. A look zanamuhla, indlela eyinhloko ukuxazulula zibalo quadratic (ngokusebenzisa discriminant) yasungulwa kuphela ekhulwini le-17 ngokusebenzisa imisebenzi Descartes futhi Newton.

Phakathi sezibalo Swiss kwekhulu le-18 Gabriel Cramer wathola indlela entsha yokwenza ikhambi izinhlelo of zibalo lwento kulula. Le ndlela iye kamuva ngegama lakhe, futhi kuze kube namuhla esiyisebenzisa ngayo. Kodwa indlela yenkulumo Kramer sika ngemva kwesikhashana, kodwa manje sizoxoxa zibalo lwento nezisombululo zabo ngokuhlukana ohlelweni.

zibalo lwento

zibalo lwento - ezothando elula nge variable (s). Basonta Aljibhra. zibalo lwento yalotshwa njengezingoma zohlobo oluthile jikelele ngendlela elandelayo: a ₁ * x + _{1 2 *} x ₂ + ... futhi _n * x _n = b. Ukulethwa leli fomu sizodinga ekulungiseleleni izinhlelo kanye matrices ku.

A system of lwento zibalo algebraic

Incazelo leli gama ithi: iqoqo zibalo ukuthi babe angaziwa ezivamile futhi ikhambi jikelele. Ngokuvamile, esikoleni zonke ixazululwe uhlelo nge zibalo ezimbili noma ezintathu. Kodwa zikhona izinhlelo nge izingxenye ezine noma ngaphezulu. Ake sibone kuqala ukuthi ukuze ukubhale phansi ukuze kamuva kwaba ezikahle ukuxazulula. Okokuqala, lesi simiso komugqa zibalo algebraic iyobukeka kahle uma wonke eziguquguqukayo abhalwe njengamazwi x ne Inkomba okuhambisanayo: 1,2,3 nokunye. Okwesibili, kufanele kuholele zonke zibalo efomini okukakhenoni: ₁ * x + _{1 2 *} x ₂ + ... futhi _n * x _n = b.

Phela lezi zinyathelo, singaqala ukukutshela indlela yokuthola ikhambi izinhlelo of zibalo lwento. Kakhulu ukuthi kuyofika ku-matrix Ukugunda.

-matrix

Matrix - itafula ukuthi siqukethe lemigqa namakholomu, futhi izakhi kwakhona at empambana yabo. Lokhu kungaba ukubaluleka ethize noma zamatshe. Ezimweni eziningi, lapho kubhekiselwa izakhi ukuthi ahlelwe ngaphansi subscripts (isib kahle ₁₁ noma _23). Inkomba yokuqala ebonisa inombolo irowu, kanti eyesibili - ikholomu. matrices Ngenhla njengoba ngenhla kanye nanoma yimaphi amanye isici zezibalo ongenza imisebenzi ehlukahlukene. Ngakho, unga:

1) khipha engeza ubukhulu efanayo etafuleni.

2) Phindaphinda matrix kunoma iyiphi inombolo noma le-vector.

3) Transpose: uguqule-matrix imigqa e amakholomu, futhi amakholomu - kumugqa.

4) Phindaphinda matrix, uma inani lemigqa ilingana omunye wabo inombolo ehlukile lamakholomu.

Ukuze uxoxe ngokuningiliziwe konke lokhu amasu, njengoba ziwusizo kithi esikhathini esizayo. Ukukhupha kanye kwalokho-matrices silula. Njengoba sithatha usayizi ofanayo-matrix, elementi etafuleni elilodwa ngayinye ihlobene wonke amanye isici. Ngakho sifaka (ukhupha) ezimbili kulezi zingxenye (kubalulekile ukuthi babemi phansi ofanayo matrices zabo). Lapho iphindaphindwe ngenani-matrix noma Vector nje wena phindaphinda elementi matrix ngalinye ukuthi inombolo (noma vector). Transposition - inqubo ethakazelisa kakhulu. Very interesting ngezinye izikhathi ukumbona empilweni yangempela, isibonelo, lapho beshintsha ukunyakaza kwezingalo a kuthebulethi yakho noma ifoni. I imifanekiso kudeskithophu kuyinto matrix, futhi ushintsho isikhundla, it is transposed futhi liya livuleka njengoba zikhula, kodwa kunciphisa ukuphakama.

Ake sihlole kabanzi inqubo efana matrix. Nakuba Yasibikela, futhi akusizi, kodwa uqaphele ukuthi namanje ewusizo. Phindaphinda matrices amabili kungaba kuphela ngaphansi kombandela wokuthi inani amakholomu itafula eyodwa ilingana nenani lemigqa nezinye. Manje thatha izakhi eyodwa-matrix umugqa nezinye izakhi kwekholomu okuhambisanayo. Phindaphinda kwamanye bese isamba ngasinye (isb, isibonelo, umkhiqizo izakhi _{11 no-12} futhi _ngo-12 b ₂₂ b kuyoba ulingana: a * b _{11 12} + ₁₂ * b futhi _22). Ngakho, itafula into eyodwa, futhi indlela efana it is eminye egcwele.

Manje sesingaqala ukucabangela indlela yokuxazulula izinhlelo of zibalo lwento.

Gauss

Le timu waqala ukuthatha indawo esikoleni. Siyazi kahle umqondo wegama elithi "uhlelo of zibalo ezimbili komugqa" futhi sazi kanjani ukuzixazulula. Kodwa kuthiwani uma inani zibalo mkhulu kunezinhliziyo ezimbili? Lokhu kuyosisiza Gauss indlela.

Yiqiniso, le ndlela kuyinto elula ukusebenzisa, uma wenza matrix kohlelo. Kodwa awukwazi ukuguqula bese unquma kwayona.

Ngakho, indlela yokuxazulula uhlelo of zibalo lwento Gauss ke? By the way, yize le ndlela futhi ngegama lakhe, kodwa wathola ke ezikhathini zasendulo. Gauss ine ukusebenza kwenziwe nge zibalo, ukuze ekugcineni kuholele isiqalo nesigcino efomini Echelon. Lokho, udinga phezulu (uma kahle beka) kusukela ngosuku lokuqala kuya equation zokugcina kwancipha ayikhulumi ngamakhosi angekho. Ngamanye amazwi, kudingeka siqiniseke ukuthi sinezinto, uthi, zibalo ezintathu: eyokuqala - angaziwa ezintathu, okwesibili - ezimbili eyesithathu - eyodwa. Khona-ke, kusukela equation zokugcina, sithola engaziwa lokuqala, bamelele ngentengo yaso yesibili noma ezothando lokuqala, futhi eminye ukuthola asele eziguquguqukayo ezimbili.

umthetho Cramer sika

Ngokuba ukuthuthukiswa yaleli su kubalulekile ukuze balukhulume kahle amakhono wokuhlanganisa, ukukhupha-matrices, kanye nesidingo ukwazi ukuthola etintfweni. Ngakho-ke, uma ungakhululekile ukwenza konke lokhu noma ungazi kanjani, kubalulekile ukufunda futhi baqeqeshwe.

Kuyini lokhu indlela, futhi kanjani ukwenza kanjalo, ukuze uthole uhlelo of zibalo lwento Cramer? Kulula kabi. Sidinga ukwakha matrix izinombolo (cishe njalo) le okuza kwesimiso komugqa zibalo Aljibhra. Ukuze wenze lokhu, bamane bathathe inombolo engaziwa, futhi sahlela isithebe ukuze alotshwe ohlelweni. Uma ngaphambi inombolo kuwuphawu "-", bese sibhala Coefficient ezimbi. Ngakho, senza matrix yokuqala okuza we angaziwa, hhayi kuhlanganise nenani ngemva setshwayo (yebo, ezothando kufanele ibe ngu- ifomu okukakhenoni uma ilungelo nje inombolo, futhi kwesokunxele - zonke angaziwa ne okuza). Khona-ke kudingeka wenze matrices ezimbalwa - elilodwa variable ngayinye. Ngenxa yale njongo, in the matrix yokuqala esikhundleni ikholomu eyodwa ikholomu ngayinye izinombolo ne okuza ngemva setshwayo lokubala. Ngakho sithola matrices ezimbalwa bese uthole etintfweni yabo.

Ngemva sathola isiphawulo, kungcono encane. Sinesibopho-matrix kokuqala, futhi kukhona matrices eziningana etholakala, okuyinto zivumelana eziguquguqukayo ezahlukene. Ukuze uthole isixazululo uhlelo, thina uhlukanise determinant kwetafula okuholela ku determinant eyinhloko etafuleni. Inombolo ewumphumela ukubaluleka omunye zamatshe. Ngokufanayo, sithola yonke angaziwa.

ezinye izindlela

Kukhona izindlela eziningana ukuze athole ikhambi izinhlelo of zibalo lwento. Ngokwesibonelo, lokho okuthiwa Gauss-Jordan indlela, esetshenziselwa abawatholi amakhambi kohlelo of zibalo quadratic, futhi sihlobana ukusetshenziswa matrices. Kukhona indlela Jacobi ukuxazulula uhlelo of lwento zibalo Aljibhra. Yena kalula uvuma wonke amakhompyutha kanye isetshenziswa wekhompyutha.

Ezimweni eziyinkimbinkimbi

Ukuba yinkimbinkimbi ngokuvamile kwenzeka uma inani zibalo kuyinto ngaphansi kwenani lokwehlukile. Khona-ke singasinda ngokuqinisekile ukuthi, noma uhlelo engahambisani (ie, ayinakho izimpande), noma inombolo izinqumo zayo ivame infinity. Uma sinalo esimweni sesibili - kubalulekile ukubhala isixazululo jikelele kohlelo of zibalo lwento. Kuyoba abandakanye okungenani umuntu oyedwa onhlobonhlobo.

isiphetho

Nanku ukuphela. Ngamafuphi: kumelwe sikuqonde lokho matrix uhlelo, ngafunda ukuthola isixazululo jikelele uhlelo of zibalo lwento. Ngaphezu kwalokho sixoxe ezinye izinketho. Thina esiphethweni indlela yokuxazulula izinhlelo of zibalo lwento: ukuqedwa Gaussian futhi umthetho Cramer sika. Saxoxa izimo ezinzima futhi nezinye izindlela ukuthola izixazululo.

Eqinisweni, lolu daba lisabalele okuningi, futhi uma ufuna ukuba uqonde kangcono ke, thina elikweluleka ufunde kabanzi izincwadi ezikhethekile.