A dongosolo la maikwezhoni liniya algebraic. Homogeneous dongosolo la maikwezhoni liniya algebraic

Kusukulu, aliyense wa ife anaphunzira aone kuti pali ndipo, ndithudi, kachitidwe ka maikwezhoni. Koma anthu ambiri sadziwa kuti pali njira zambiri kuwathetsa. Lero tiona ndendende onse njira zothetsera dongosolo la maikwezhoni liniya algebraic, amene wapangidwa maikwezhoni oposa awiri.

nkhani

Lero tikudziwa kuti luso kuthetsa maikwezhoni ndi kachitidwe kawo unachokera mu Babulo ndi ku Iguputo. Komabe, kufanana mu mawonekedwe awo bwino anaonekera kwa ife pambuyo pakutchulidwa wa chizindikiro wofanana "=", umene unayamba mu 1556 ndi umboni English masamu. Mwa njira, chizindikiro ichi anasankhidwa chifukwa: zikutanthauza awiri kufanana zigawo ofanana. Zoonadi, chitsanzo chabwino ofanana Kodi sanabwere.

Woyambitsa wa lettering ano ndi zizindikiro za mmene zosadziwika, French masamu Fransua việt. Komabe, akuti ake n'chosiyana kwambiri lero. Mwachitsanzo, lalikulu la nambala yosadziŵika iye zinaperekedwa mwa kalata Q (balati "quadratus".), Ndipo kyubu - (. Balati "cubus") kalata C. zizindikiro izi tsopano ngati wovuta, koma ndiye njira kwambiri mwachilengedwe kulemba dongosolo la maikwezhoni liniya algebraic.

Komabe, sangathe mu njira ofala njira chinali masamu takambiranazi yekha mizu positive. Mwina zimenezi chifukwa chakuti makhalidwe oipa mulibe ntchito iliyonse othandiza. mwanjira ina, koma woyamba kuonedwa kuti mizu zoipa linayamba masamu Italy Niccolo Tartaglia, Gerolamo Cardano ndi Raphael Bombelli m'nthawi ya 16. Kuyang'ana ano, njira yaikulu kuthetseratu maikwezhoni quadratic (kudzera discriminant) inakhazikitsidwa ndi m'zaka za m'ma 17 mwa ntchito za Descartes ndi Newton.

Mu zaka za m'ma 18 Swiss masamu Gabriel Cramer kupeza njira zatsopano ndi njira ya machitidwe a maikwezhoni liniya mosavuta. Njira patapita dzina lake, ndipo lero ife ntchito. Koma pa njira ya nkhani Kramer a mtsogolo pang'ono, koma tsopano tikambirana maikwezhoni liniya ndi njira zawo mosiyana dongosolo.

maikwezhoni liniya

Liniya maikwezhoni - anthu aone kuti pali losavuta ndi variable (s). Iwo ali algebraic lapansi. maikwezhoni liniya olembedwa mu mawonekedwe ambiri motere: ₁ * × ₁ + _{2 *} × ₂ + ... ndipo _N * × _N = b. Kugonjera fomu iyi tiyenera pokonza machitidwe ndi mawerengetsedwe pa.

A dongosolo la maikwezhoni liniya algebraic

The tanthauzo la mawu amenewa ndi: akonzedwa a maikwezhoni kuti ndi unknowns wamba ndi njira ambiri. Childs, kusukulu onse anathana dongosolo ndi maikwezhoni awiri kapena atatu. Koma pali machitidwe ndi zigawo zinayi kapena kuposa. Tiyeni tiwone woyamba momwe usazilembe kuti patapita anali yabwino yothetsera. Choyamba, kachitidwe ka maikwezhoni liniya algebraic adzayang'ana bwino ngati zosintha zonse zinalembedwa monga × ndi index lolingana: 1,2,3 ndi zina zotero. Kachiwiri, ayenera kutsogolera maikwezhoni zonse mawonekedwe ovomerezeka: ₁ * × ₁ + _{2 *} × ₂ + ... ndipo _N * × _N = b.

Pambuyo masitepe awa onse, Tikhoza kukuuzani inu momwe kupeza yankho la machitidwe a maikwezhoni liniya. Kwambiri kwa adzabwera imathandiza masanjidwewo.

masanjidwewo

Masanjidwewo - tebulo kuti tichipeza mizere ndi mizati, ndi zinthu zake pa mphambano zawo. Zimenezi zingakhale kapena mtengo kapena variable. Nthawi zambiri, kuti uzipereke zinthu zimene imakonzedwa pansi subscripts ndi (mwachitsanzo, ndi ₁₁ kapena ₂₃ bwino). The index woyamba limasonyeza mzere nambala, ndipo chachiwiri - ndime. mawerengetsedwe pamwamba monga pamwamba ndi zina mchitidwe masamu angachichite ntchito zosiyanasiyana. Motero, mungathe:

1) kuchotsera ndipo kuwonjezera kukula yomweyo pa tebulo.

2) awonjezere masanjidwewo kwa aliyense kapena vekitala.

3) Transpose: kusandutsa masanjidwewo malire mizati, ndi mizati - mu mzere.

4) Chulukitsani ndi masanjidwewo, ngati chiwerengero cha mizere ndi wofanana ndi wa iwo chiwerengero osiyanasiyana m'manyuzipepala.

Kukambirana mwatsatanetsatane zonse njira imeneyi, monga iwo akutigwiritsa ntchito m'tsogolo. Wochotsera ndi kuonjedzera wa mawerengetsedwe lophweka. Popeza ife timatenga yemweyo kukula masanjidwewo, aliyense mchitidwe tebulo limodzi ndi ofanana ndi zinthu zonse zina. Choncho ife kuwonjezera (kuchotsera) awiri a zinthu izi (nkofunika kuti iwo anali kuyima pa nthaka yomweyi mu mawerengetsedwe awo). Pamene kuchulukitsa ndi chiwerengero cha masanjidwewo kapena vekitala chabe kuchulukitsa lililonse mchitidwe masanjidwewo ndi chiwerengero (kapena vekitala). Transposition - njira chidwi kwambiri. chidwi kwambiri nthawi zina kumuona mu moyo weniweniwo Mwachitsanzo, pamene kusintha lathu la piritsi kapena foni. The zithunzi pa kompyuta ndi bokosi, ndi kusintha kwa malo, ndi transposed ndipo onse, koma amachepetsa mu msinkhu.

Tiyeni tione zina zimachitika monga masanjidwewo chulutsa. Ngakhale adatiwuza ife, ndipo si zothandiza, koma kudziwa adakali zothandiza. Chulukanani mawerengetsedwe awiri kungakhale kokha pansi chikhalidwe chimene chiwerengero cha mizati mu tebulo limodzi ndi wofanana ndi chiwerengero cha mizere ena. Tsopano kutenga zinthu limodzi masanjidwewo mzere ndi zinthu zina za ndime chogwirizana. Chulukitsani iwo wina ndi mzake ndipo kenako ndalama (i.e. Mwachitsanzo, mankhwala a zinthu ₁₁ ndi ₁₂ ndipo pa ₁₂ B ndi ₂₂ b adzakhala wofanana: a * b _{11 12} + ₁₂ * B ndi _22). Choncho, limodzi patebulo item, ndi njira yofanana ndi iyo ladzala zina.

Tsopano Tikhoza kuganizira mmene angathetsere machitidwe a maikwezhoni liniya.

Gauss

Izi lathu anayamba kutenga malo pa sukulu. Ife tikudziwa bwino mfundo yonena za "dongosolo la maikwezhoni awiri liniya" ndipo mudziwa kuwathetsa. Koma bwanji ngati chiwerengero cha maikwezhoni ndi wamkulu kuposa awiri? Itithandiza Gauss njira.

Kumene, njira imeneyi ndi yabwino kwa ntchito, ngati muli bokosi la kachitidwe. Koma simungathe atembenuke ndi kusankha yake.

Kotero, momwe kulikonza ilo ndi dongosolo la liniya maikwezhoni Gauss? Mwa njira, ngakhale njira imeneyi ndi dzina lake, koma anapeza kuti kalelo. Gauss ali opareshoni ikuchitika ndi maikwezhoni, kuti potsirizira pake chikachititsa kuchotsedwa kwa mtundu echelon. Ndiko kuti, muyenera pamwamba-pansi (ngati molondola ikani) koyambirira kwa aone kuti pali otsiriza chitachepa amodzi osadziwika. M'mawu ena, tiyenera kuonetsetsa kuti ife tiri, kunena, maikwezhoni atatu: choyamba - unknowns atatu, lachiwiri - awiri wachitatu - wina. Ndiye, kuchokera aone kuti pali lotsiriza, ife tikupeza zosadziwika choyamba, mmalo phindu lake lachiwiri kapena aone kuti pali woyamba, ndi zina otsala zosintha ziwiri.

ulamuliro Cramer a

Pa chitukuko cha njira imeneyi n'kofunika kuti adziwe maluso a Komanso, wochotsera wa mawerengetsedwe, komanso kufunika kupeza determinants. Choncho, ngati ndinu wovuta kuchita izi zonse kapena sindikudziwa momwe, m'pofunika kuphunzira ndi kuziphunzitsa.

ndicho njira yotani, ndipo kutero, kuti dongosolo la liniya maikwezhoni Cramer? Ndi yosavuta. Tiyenera kumanga bokosi la manambala (nthawi) ndi koefishienti a dongosolo la maikwezhoni liniya algebraic. Kuti tichite zimenezi, amangozindikira chiwerengero cha osadziwika, ndipo ife kukonza tebulo mu dongosolo kuti iwo analembedwa mu dongosolo. Ngati pamaso chiwerengero ndi chizindikiro "-", ndiye ife kulemba koyefishienti negative. Kotero, ife anapanga masanjidwewo loyamba la koefishienti wa unknowns, osati kuphatikizapo chiwerengero pambuyo chizindikiro ofanana (kumene, kuti aone kuti pali ayenera kuchepetsedwa kwa mtundu wodziwika pamene lamanja basi angapo, ndi lamanzere - ndi unknowns onse ndi koefishienti). Ndiye muyenera kupanga mawerengetsedwe ochepa - aliyense variable. Chifukwa chaichi, mu masanjidwewo choyamba m'malo mwa ndime imodzi manambala pagawo lililonse makamaka ndi koefishienti pambuyo chizindikiro ofanana. Choncho ife mawerengetsedwe ochepa kenako kupeza determinants awo.

Tikapeza ndi qualifiers, ndi ang'ono. Tili ndi masanjidwewo koyamba, ndipo pali mawerengetsedwe angapo anachokera, omwe akufanana ndi zosintha osiyana. Kuti dongosolo njira, ife kugawanitsa determinant ku gome chifukwa pa determinant chachikulu pa tebulo. Chiwerengero chifukwa ndicho phindu la variable wina. Mofananamo, ife tikupeza unknowns onse.

njira zina

Pali njira zosiyanasiyana kuti mupeze yankho la machitidwe a maikwezhoni liniya. Mwachitsanzo, otchedwa Gauss-Jordan njira, amene ntchito kupeza njira ya mapeto a maikwezhoni quadratic, komanso anafotokoza kuti ntchito mawerengetsedwe. Palinso njira Jacobi kwa kuthetsa dongosolo la maikwezhoni liniya algebraic. Iye mosavuta Amasintha makompyuta onse ndi ntchito kompyuta.

Nthawi yovuta

Kumvetsa zambiri kumachitika ngati chiwerengero cha maikwezhoni ndi zosakwana chiwerengero cha zosintha. Ndiye tikhoza ndithu kuti, kapena kachitidwe losemphana (ie, alibe mizu), kapena chiwerengero cha zosankha zake amamuchititsa osawerengeka. Ngati tili ndi vuto chachiwiri - m'pofunika kulemba yankho ambiri a dongosolo la maikwezhoni liniya. Chidzaphatikiza variable chimodzi.

Pomaliza

Apa ife tikubwera ku mapeto. Kuti adzimvera: tiyenera kumvetsa zimene dongosolo masanjidwewo, anaphunzira kupeza yankho ambiri a dongosolo la maikwezhoni liniya. Komanso tinakambirana zina zimene mungachite. Tidadzatulukira mmene angathetsere machitidwe a maikwezhoni liniya: ya Gauss Kupha ndi ulamuliro Cramer a. Tinakambirana za milandu yovuta ndi njira zina za kupeza njira zothetsera mavutowo.

Ndipotu nkhani zambiri ochulukirapo, ndipo ngati mukufuna kumvetsa bwino izo, ife inu kuti muwerenge zambiri za mabuku apadera a.