Ib tug system ntawm linear algebraic equations. Homogeneous system ntawm linear equations algebraic

Thaum lub tsev kawm ntawv, peb txhua tus kawm txog kab zauv thiab, yeej, cov kev sib npaug. Tab sis tsis muaj coob tus neeg paub hais tias muaj ntau txoj kev uas yuav daws tau lawv. Niaj hnub no peb yuav pom raws nraim li tag nrho cov kev daws ib tug system ntawm linear algebraic equations, uas yog tsim los ntawm ntau tshaj li ob npaug.

zaj dabneeg

Niaj hnub no peb paub tias lub kos duab ntawm kev daws sib npaug thiab lawv lub nruab originated nyob rau hauv ancient Babylon thiab tim lyiv teb chaws. Txawm li cas los, koob pheej ntawm lawv nyob rau hauv lawv paub daim ntawv tshwm rau peb tom qab qhov tshwm sim ntawm cov sib npaug zos kos npe rau "=", uas yog tswvcuab hauv 1556 los ntawm lus Askiv mathematician cov ntaub ntawv. Los ntawm txoj kev, lub cim raug xaiv rau ib tug yog vim li cas: nws txhais tau tias ob tug uas tig mus sib npaug zos feem. Tseeb, qhov zoo tshaj plaws piv txwv ntawm koob pheej ntawm lawv tsis tuaj.

Qhov tus tsim ntawm niaj hnub lettering thiab cov cim ntawm tsis paub hais tias qhov twg, lub Fabkis mathematician Fransua Viet. Txawm li cas los, nws los yog ho sib txawv ntawm hnub no. Piv txwv li, ib tug square ntawm ib tug tsis paub tus naj npawb nws tso cai los ntawm tsab ntawv Q (lat "quadratus".), Thiab cov thawv muaj rau sab - tsab ntawv C (lat "cubus".). Cov cim tam sim no zoo li tsis xis nyob, tab sis, ces nws yog tus tshaj plaws intuitive txoj kev uas yuav sau ib qhov system of linear algebraic equations.

Txawm li cas los, ib tug disadvantage nyob rau hauv lub prevailing txoj kev tshuaj yog tias daim ntawv thom nug tau kawm xwb qhov zoo cov hauv paus hniav. Tej zaum qhov no yog vim lub fact tias tsis zoo qhov tseem ceeb tsis muaj tswv yim daim ntawv thov. Ib txoj kev los sis lwm, tab sis tus thawj yuav pom tau tias tsis zoo hauv paus hniav pib tom qab lub Italian lej Niccolo Tartaglia, Gerolamo Cardano thiab Raphael Bombelli nyob rau hauv lub xyoo pua 16th. Ib tug niaj hnub saib, lub ntsiab txoj kev daws quadratic equations (los ntawm discriminant) yog tsim los pab xwb nyob rau hauv lub xyoo pua 17th los ntawm tej hauj lwm ntawm Descartes thiab Newton.

Nyob rau hauv nruab nrab ntawm lub 18th xyoo pua Swiss mathematician Gabriel Cramer pom ib tug tshiab txoj kev uas yuav ua rau cov tshuaj ntawm lub ntawm linear equations yooj yim. Qhov no txoj kev tom qab tis npe hu ua tom qab nws, thiab hnub no peb siv nws. Tab sis nyob rau cov qauv ntawm cov Kramer tus tham ib tug me ntsis rau yav tom ntej, tab sis rau tam sim no peb yuav sib tham txog linear equations thiab lawv cov kev daws nyias los ntawm lub system.

linear equations

Tawm sib npaug - qhov nyuaj kab zauv nrog nce mus nce los (s). Lawv mus rau lub algebraic. Tawm sib npaug sau nyob rau hauv lub dav dav daim ntawv raws li nram no: ₁ * x ₁ + _{2 *} x ₂ + ... thiab _n * x _n = b. Kev mloog lus ntawm daim ntawv no peb yuav tsum tau nyob rau hauv qhov kev npaj ntawm systems thiab matrices rau.

Ib tug system ntawm linear equations algebraic

Lub ntsiab txhais ntawm no lub sij hawm yog: ib tug txheej ntawm equations uas muaj ib qho unknowns thiab cov tshuaj. Feem ntau, tom tsev kawm ntawv tag nrho solved ib tug system nrog ob los yog peb npaug. Tab sis muaj nruab nrog plaub los yog ntau tshaj Cheebtsam. Cia peb kawm saib ua ntej yuav ua li cas sau ntawv cia kom hais tias tom qab nws yog yooj yim los mus daws kom tau. Firstly, lub system ntawm linear algebraic equations yuav saib zoo dua yog tias tag nrho cov tsiaj ntawv no yog sau raws li x nrog cov coj index: 1,2,3 thiab thiaj li nyob. Secondly, nws yuav tsum ua tag nrho cov sib npaug mus rau lub canonical daim ntawv: ₁ * x ₁ + _{2 *} x ₂ + ... thiab _n * x _n = b.

Tom qab tag nrho cov kauj ruam, peb yuav tau pib qhia rau koj yuav ua li cas nrhiav tau cov tshuaj ntawm lub ntawm linear equations. Heev npaum li cas rau cov uas yuav tuaj nyob rau hauv muab rau matrix.

matrix

Matrix - ib lub rooj uas muaj kab thiab txhua, thiab nws cov ntsiab yog nyob rau ntawm lawv kev sib tshuam. Qhov no yuav tsum yog ib qho kev muaj nuj nqis los yog nce mus nce los. Nyob rau hauv feem ntau tus neeg mob, seb yuav muab cov ntsiab uas cov txheej txheem hauv qab lub subscripts (e.g., ib tug ₁₁ los yog ₂₃ zoo). Tus thawj index qhia cov kab xov tooj, thiab lub thib ob - sab. Saum toj no matrices li saum toj no thiab lwm yam xyuam xim lub caij yuav ua tau ntau yam hauj lwm. Yog li, koj yuav:

1) Rho thiab ntxiv tus loj tib yam ntawm lub rooj.

2) Muab lub matrix rau tej xov tooj los yog vector.

3) Transpose: txia matrix kab nyob rau hauv lub kab, thiab cov kab - nyob rau hauv kab.

4) Muab lub matrix, yog hais tias tus naj npawb ntawm cov kab yog sib npaug zos rau ib tug ntawm lawv ib tug txawv tus naj npawb ntawm kab.

Yuav kom sib tham txog nyob rau hauv kom meej tag nrho cov hom kev kawm, raws li lawv yog pab tau rau peb nyob rau hauv lub neej yav tom ntej. Rho thiab tsis tas li ntawd ntawm matrices yog heev yooj yim. Txij li thaum peb noj tib yam loj matrix, txhua lub caij ntawm ib lub rooj yog lwm yam kev mus rau txhua txhua lwm yam caij. Yog li peb ntxiv (rho) ob ntawm cov ntsiab (nws yog ib qho tseem ceeb hais tias lawv sawv nyob rau hauv tib av nyob rau hauv lawv cov matrices). Thaum khoo los ntawm cov xov tooj ntawm matrix los yog vector koj tsuas muab txhua lub caij ntawm lub matrix los ntawm tus xov tooj (los yog vector). Transposition - ib tug heev interesting txheej txheem. Heev ntawm nthuav cov tej zaum yuav pom nws nyob rau hauv lub neej tiag tiag, piv txwv li, thaum hloov lub orientation ntawm ib ntsiav tshuaj los yog xov tooj. Cov icons rau lub desktop yog ib tug matrix, thiab nrog ib tug kev hloov ntawm txoj hauj lwm, nws yog transposed thiab yuav dav, tab sis txo nyob rau hauv qhov siab.

Cia peb kawm ntxiv ib tug txheej txheem xws li matrix npaug. Txawm tias nws hais rau peb, thiab yog tsis pab tau, tab sis yuav tsum paub nws yog tseem pab tau. Muab ob matrices yuav ua tau tsuas nyob rau hauv cov mob uas lub xov tooj ntawm kab nyob rau hauv ib lub rooj yog sib npaug zos rau cov xov tooj ntawm kab lwm yam. Tam sim no siv ib matrix kab ntsiab thiab lwm yam hais txog cov coj kiag. Muab lawv rau txhua lwm yam thiab ces zaum (i.e., piv txwv li, ib yam khoom hais ₁₁ thiab ₁₂ thiab nyob rau hauv ₁₂ b thiab ₂₂ b yuav ua ncaj rau: ib tug * b _{11 12} + ₁₂ * b thiab _22). Yog li, muaj ib zaug xwb rooj khoom, thiab ib txoj kev zoo xws li cov nws yog lawm ua tus sau ntxiv.

Tam sim no peb yuav tau pib xav txog yuav ua li cas los daws kom tau lub nruab ntawm linear equations.

Gauss

No lub ntsiab pib mus rau qhov chaw ntawm lub tsev kawm ntawv. Peb paub zoo heev txog lub tswvyim ntawm "system ntawm ob qhov linear equations" thiab paub yuav ua li cas los daws kom tau lawv. Tab sis li cas yog hais tias tus naj npawb ntawm cov equations yog ntau tshaj ob tug? Qhov no yuav pab kom peb Gauss txoj kev.

Ntawm cov hoob kawm, qhov no txoj kev yog yooj yim siv, Yog hais tias koj ua ib tug matrix ntawm lub system. Tab sis koj muaj peev xwm tsis hloov nws thiab txiav txim siab rau nws tus kheej.

Yog li ntawd, yuav ua li cas los daws kom tau nws los ntawm ib tug system ntawm linear equations Gauss? Los ntawm txoj kev, txawm tias qhov no txoj kev thiab muaj npe tom qab nws, tab sis pheej nrhiav tau nws nyob rau hauv ancient sij hawm. Gauss muaj ib lub lag luam nqa tawm nrog qhov sib npaug, kom nws thiaj li ua nyob rau hauv lub totality rau echelon daim ntawv. Uas yog, koj yuav tsum mus rau saum-down (yog tias raug muab) los ntawm tus thawj kom kawg kab zauv waned ib tsis paub hais tias. Nyob rau hauv lwm yam lus, peb yuav tsum tau kom paub tseeb tias peb twb tau txais, hais tias, peb sib npaug: thawj - peb unknowns, nyob rau hauv lub thib ob - ob nyob rau hauv lub thib peb - ib tug. Ces, los ntawm lub xeem kab zauv, peb nrhiav tau cov thawj tsis paub hais tias, tus nws tus nqi nyob rau hauv lub thib ob los yog tus thawj kab zauv, thiab ntxiv nrhiav tus seem ob ntau yam.

Cramer kav

Rau txoj kev loj hlob ntawm cov txheej txheem no yog tseem ceeb heev kom txawj txawj ntxiv, kev rho tawm ntawm matrices, raws li zoo raws li qhov yuav tsum tau yuav tsum tau mus nrhiav determinants. Yog li ntawd, yog hais tias koj tsis xis nyob ua li no tag nrho los yog tsis paub yuav ua li cas, nws yog tsim nyog los kawm thiab tau kawm.

Yuav ua li cas yog lub essence ntawm no txoj kev, thiab yuav ua li cas ua li ntawd, kom tau ib tug system ntawm linear equations Cramer? Nws yog heev yooj yim. Peb yuav tsum tau los tsim kom tau ib tug matrix ntawm tus xov tooj (yuav luag txhua zaus) lub coefficients ntawm ib tug system ntawm linear algebraic equations. Ua li no, tsuas coj tus naj npawb ntawm tsis paub hais tias, thiab peb npaj ib lub rooj nyob rau hauv qhov kev txiav txim hais tias lawv yog kaw nyob rau hauv lub system. Yog hais tias ua ntej tus xov tooj yog ib tug kos npe rau "-", ces yog peb sau ntawv tsis zoo coefficient. Yog li ntawd, peb ua tus thawj matrix ntawm lub coefficients ntawm lub unknowns, tsis xws li tus xov tooj tom qab lub sib npaug zos kos npe rau (ntawm chav kawm, tias cov kab zauv yuav tsum tau txo kom lub canonical daim ntawv thaum txoj cai yog cia li ib tug xov tooj, thiab rau sab laug - tag nrho cov unknowns nrog coefficients). Ces koj yuav tau ua ib ob peb matrices - ib qho rau ib nce mus nce los. Rau lub hom phiaj no, nyob rau hauv thawj matrix yog hloov los ntawm ib kem ib sab tus xov tooj nrog lub coefficients tom qab lub sib npaug zos kos npe rau. Yog li peb yuav tau ib tug ob peb matrices thiab ces nrhiav tau lawv determinants.

Tom qab peb pom lub qualifiers, nws yog me me. Peb muaj ib tug thawj zaug matrix, thiab muaj ntau ntau muab matrices, uas sib tau txawv zog. Yog xav tau ib tug system tov, peb tau faib cov determinant ntawm lub resulting rooj rau cov thawj determinant ntawm lub rooj. Cov uas ua tus xov tooj yog tus nqi ntawm ib tug nce mus nce los. Ib yam li ntawd, peb yuav nrhiav tau tag nrho cov unknowns.

lwm txoj kev

Muaj ob peb txoj kev nyob rau hauv thiaj li yuav tau cov tshuaj ntawm lub ntawm linear equations. Piv txwv li, ib tug thiaj li hu ua Gauss-Jordan txoj kev, uas yog siv rau kev nrhiav cov ntsiab ntawm lub system ntawm quadratic equations, thiab kuj hais txog kev siv ntawm matrices. Muaj kuj yog ib tug Jacobi txoj kev daws ib tug system ntawm linear algebraic equations. Nws yooj yim adapts rau tag nrho cov computers thiab yog siv nyob rau hauv koj laij.

nyuab mob

Complexity feem ntau tshwm sim yog hais tias tus naj npawb ntawm cov equations yog tsawg tshaj li tus xov tooj ntawm cov tsiaj ntawv. Tom qab ntawd peb yuav yeej hais tias, los yog lub system yog inconsistent (ie, muaj tsis muaj keeb kwm), los yog tus xov tooj ntawm nws cov kev txiav txim siab nyhav infinity. Yog hais tias peb muaj lub thib ob cov ntaub ntawv - nws yog tsim nyog los sau cov kev tshuaj ntawm lub system ntawm linear equations. Nws yuav muaj xws li yog tsawg kawg yog ib tug nce mus nce los.

xaus

Ntawm no yog peb tuaj mus rau thaum xaus. Yuav kom saib lub ntsab lug: peb yuav tsum to taub dab tsi lub system matrix, kawm tau mus nrhiav tau cov kev tshuaj ntawm ib tug system ntawm linear equations. Nyob rau hauv tas li ntawd peb pom tau hais tias lwm cov kev xaiv. Peb figured tawm yuav ua li cas los daws kom tau lub nruab ntawm linear equations: Gaussian tshem tawm thiab Cramer txoj cai. Peb tham txog qhov nyuaj mob thiab lwm cov kev uas nrhiav kev daws teeb meem.

Nyob rau hauv qhov tseeb, qhov no qhov teeb meem yog ntau npaum li cas uas nws kim heev, thiab yog hais tias koj xav tau kom to taub nws, peb qhia rau koj mus nyeem ntxiv ntawm cov tshwj xeeb ntaub ntawv.