U linijnij algebri dlya vektornih prostoriv V displaystyle V i W displaystyle W nad polem K displaystyle K bud yake linijne vidobrazhennya T V W displaystyle T V to W mozhna podati za dopomogoyu matrici yaka nazivayetsya matriceyu linijnogo vidobrazhennya Dane predstavlennya ye zruchnim dlya obchislen 1 ta dozvolyaye obchislyuvati kompoziciyu linijnih vidobrazhen cherez zvichajnij dobutok matric Prote matricya linijnogo vidobrazhennya viznachena ne odnoznachno a zalezhit vid viboru bazisiv u prostorah V displaystyle V i W displaystyle W Matrici linijnogo peretvorennya u riznih bazisah pov yazani matrichnoyu totozhnistyu iz vikoristannyam matric perehodu mizh riznimi bazisami Zmist 1 Oznachennya 2 Spivvidnoshennya mizh koordinatami vektoriv 3 Prikladi 3 1 Peretvorennya na ploshini 4 Vlastivosti 4 1 Izomorfizmi mizh prostorami linijnih vidobrazhen i matric 4 2 Matricya kompoziciyi linijnih vidobrazhen 4 3 Zmina matrici pri perehodi do novih bazisiv 5 Matrici deyakih nelinijnih vidobrazhen 6 Dzherela 7 PrimitkiOznachennya RedaguvatiNehaj V displaystyle V nbsp ye vektornim prostorom rozmirnosti n displaystyle n nbsp nad polem K displaystyle K nbsp iz vibranim na nomu bazisom A a 1 a n displaystyle mathcal A mathbf a 1 ldots mathbf a n nbsp a W displaystyle W nbsp ye vektornim prostorom rozmirnosti m displaystyle m nbsp nad polem K displaystyle K nbsp iz vibranim bazisom B b 1 b m displaystyle mathcal B mathbf b 1 ldots mathbf b m nbsp Nehaj T V W displaystyle T V to W nbsp ye linijnim vidobrazhennyam mizh cimi dvoma prostorami Matriceyu linijnogo vidobrazhennya T displaystyle T nbsp u bazisah A displaystyle mathcal A nbsp i B displaystyle mathcal B nbsp poznachatimetsya M T A B displaystyle M T mathcal A mathcal B nbsp nazivayetsya matricya stovpcyami yakoyi ye koeficiyenti rozkladu vektoriv T a j displaystyle T mathbf a j nbsp tobto obraziv vektoriv bazisu A displaystyle mathcal A nbsp u bazisi B displaystyle mathcal B nbsp Bilsh detalno kozhen vektor T a j W displaystyle T mathbf a j in W nbsp mozhna u yedinij sposib zapisati cherez elementi B displaystyle mathcal B nbsp viznachivshi koeficiyenti a i j displaystyle a ij nbsp T a j a 1 j b 1 a 2 j b 2 a m j b m i 1 m a i j b i displaystyle T mathbf a j a 1j mathbf b 1 a 2j mathbf b 2 dotsb a mj mathbf b m sum i 1 m a ij mathbf b i nbsp Todi matricya linijnogo vidobrazhennya u cih bazisah matime viglyad M T A B a 11 a 1 j a 1 n a 21 a 2 j a 2 n a m 1 a m j a m n displaystyle M T mathcal A mathcal B begin pmatrix a 11 amp dots amp a 1j amp dots amp a 1n a 21 amp dots amp a 2j amp dots amp a 2n vdots amp amp vdots amp amp vdots a m1 amp dots amp a mj amp dots amp a mn end pmatrix nbsp Zokrema yaksho V displaystyle V nbsp ye prostorom rozmirnosti n displaystyle n nbsp a W displaystyle W nbsp ye prostorom rozmirnosti m displaystyle m nbsp to matricya dovilnogo linijnogo vidobrazhennya dlya dovilnih bazisiv matime poryadok m n displaystyle m times n nbsp Yaksho V W displaystyle V W nbsp todi linijne vidobrazhennya perevazhno nazivayetsya linijnim peretvorennyam to matricya dovilnogo linijnogo peretvorennya ye kvadratnoyu Yaksho u comu vipadku bazisi A displaystyle mathcal A nbsp i B displaystyle mathcal B nbsp ye odnakovimi dlya ciyeyi matrici vikoristovuyetsya poznachennya M T A displaystyle M T mathcal A nbsp Zauvazhennya Podane tut oznachennya ye najposhirenishim u literaturi ale inodi mozhe vikoristovuvatisya oznachennya de koeficiyenti rozkladu vektoriv T a j displaystyle T mathbf a j nbsp u bazisi prostoru W displaystyle W nbsp utvoryuyut ryadki a ne stovpci matrici Spivvidnoshennya mizh koordinatami vektoriv RedaguvatiNehaj vektorni prostori V displaystyle V nbsp i W displaystyle W nbsp yih bazisi A a 1 a n displaystyle mathcal A mathbf a 1 ldots mathbf a n nbsp i B b 1 b n displaystyle mathcal B mathbf b 1 ldots mathbf b n nbsp i linijne vidobrazhennya T V W displaystyle T V to W nbsp zadani yak i vishe Kozhen vektor x V displaystyle mathbf x in V nbsp mozhna odnoznachno zapisati cherez elementi bazisa A displaystyle mathcal A nbsp x x 1 a 1 x n a n displaystyle mathbf x x 1 mathbf a 1 dotsb x n mathbf a n nbsp Analogichno T x W displaystyle T mathbf x in W nbsp mozhna odnoznachno zapisati cherez elementi bazisa B displaystyle mathcal B nbsp T x y 1 b 1 y n b n displaystyle T mathbf x y 1 mathbf b 1 dotsb y n mathbf b n nbsp Oderzhani takim chinom koordinati mozhna zapisati yak vektor stovpci x 1 x n displaystyle begin pmatrix x 1 vdots x n end pmatrix nbsp i y 1 y m displaystyle begin pmatrix y 1 vdots y m end pmatrix nbsp Todi ci koordinati pov yazani mizh soboyu cherez matricyu linijnogo peretvorennya M T A B displaystyle M T mathcal A mathcal B nbsp y 1 y m a 11 a 1 n a m 1 a m n x 1 x n displaystyle begin pmatrix y 1 vdots y m end pmatrix begin pmatrix a 11 amp dots amp a 1n vdots amp ddots amp vdots a m1 amp dots amp a mn end pmatrix cdot begin pmatrix x 1 vdots x n end pmatrix nbsp Prikladi RedaguvatiMatricya totozhnogo peretvorennya tobto V W displaystyle V W nbsp i T x x x V displaystyle T x x forall x in V nbsp u vipadku yaksho bazisi A displaystyle mathcal A nbsp i B displaystyle mathcal B nbsp ye odnakovimi ye odinichnoyu matriceyu Yaksho natomist vibrati rizni bazisi to cya matricya bude rivnoyu matrici perehodu mizh bazisami Nehaj P n x displaystyle P n x nbsp poznachaye linijnij prostir mnogochleniv stepeni yakih ne perevishuyut n displaystyle n nbsp Nehaj na kozhnomu takomu prostori vibrano standartnij bazis 1 x 1 x 2 x n displaystyle 1 x 1 x 2 ldots x n nbsp Operator formalnogo diferenciyuvannya a 0 a 1 x 1 a 2 x 2 a n x n a 1 2 a 2 x 1 n a n x n 1 displaystyle a 0 a 1 x 1 a 2 x 2 ldots a n x n a 1 2a 2 x 1 ldots na n x n 1 nbsp ye linijnim operatorom iz P n x displaystyle P n x nbsp u P n 1 x displaystyle P n 1 x nbsp iz standartnimi bazami Matriceyu cogo peretvorennya ye n n 1 displaystyle n times n 1 nbsp matricya vidu 0 1 0 0 0 0 0 2 0 0 0 0 0 3 0 0 0 0 0 n displaystyle begin pmatrix 0 amp 1 amp 0 amp 0 amp dots amp 0 0 amp 0 amp 2 amp 0 amp dots amp 0 0 amp 0 amp 0 amp 3 amp dots amp 0 vdots amp vdots amp vdots amp vdots amp ddots amp vdots 0 amp 0 amp 0 amp 0 amp dots amp n end pmatrix nbsp dd Peretvorennya na ploshini Redaguvati Vsyudi nizhche vikoristovuyetsya yedinij bazis dlya koordinat obrazu i proobrazu peretvorennya ObertannyaFunkcionalna forma zapisu obertannya na kut 8 proti godinnikovoyi strilki vidnosno pochatku koordinat x x cos 8 y sin 8 y x sin 8 y cos 8 displaystyle begin cases x x cos theta y sin theta y x sin theta y cos theta end cases nbsp dd Tobto vektor iz koordinatami x y displaystyle x y nbsp perehodit u vektor iz koordinatami x y displaystyle x y nbsp Matriceyu cogo linijnogo peretvorennya ye matricya povorotu i u matrichnij formi mozhna zapisati x y cos 8 sin 8 sin 8 cos 8 x y displaystyle begin pmatrix x y end pmatrix begin pmatrix cos theta amp sin theta sin theta amp cos theta end pmatrix begin pmatrix x y end pmatrix nbsp dd MasshtabuvannyaFunkcionalna forma masshtabuvannya x s x x y s y y displaystyle begin cases x s x cdot x y s y cdot y end cases nbsp dd Matriceyu cogo peretvorennya ye diagonalna matricya x y s x 0 0 s y x y displaystyle begin pmatrix x y end pmatrix begin pmatrix s x amp 0 0 amp s y end pmatrix begin pmatrix x y end pmatrix nbsp dd Koli s x s y 1 displaystyle s x s y 1 nbsp todi zberigayetsya plosha ZsuvU vipadku zsuvu shear mozhlivi dva varianti Zsuv po osi x x x k y displaystyle x x ky nbsp i y y displaystyle y y nbsp todi matricya zsuvu maye viglyad x y 1 k 0 1 x y displaystyle begin pmatrix x y end pmatrix begin pmatrix 1 amp k 0 amp 1 end pmatrix begin pmatrix x y end pmatrix nbsp dd Zsuv po osi y x x displaystyle x x nbsp and y y k x displaystyle y y kx nbsp v comu vipadku x y 1 0 k 1 x y displaystyle begin pmatrix x y end pmatrix begin pmatrix 1 amp 0 k amp 1 end pmatrix begin pmatrix x y end pmatrix nbsp dd VidbittyaDlya vidbittya vektora shodo pryamoyi yaka prohodit cherez pochatok koordinat nehaj lx ly vektor sho lezhit na pryamij Matriceyu vidbittya shodo ciyeyi pryamoyi ye matricya Hausholdera A 1 l x 2 l y 2 l x 2 l y 2 2 l x l y 2 l x l y l y 2 l x 2 displaystyle mathbf A frac 1 l x 2 l y 2 begin pmatrix l x 2 l y 2 amp 2l x l y 2l x l y amp l y 2 l x 2 end pmatrix nbsp dd Vidbittya vidnosno pryamoyi yaka ne prohodit cherez pochatok koordinat ne ye linijnim peretvorennyam ce peretvorennya afinne Dlya vidbittya tochki vidnosno ploshini a x b y c z 0 displaystyle ax by cz 0 nbsp mozhna vikoristati rivnyannya I 2 N N T displaystyle I 2NN T nbsp de I odinichna matricya i N odinichnij vektor normali do ploshini Matricya peretvorennya bude mati viglyad 1 2 a 2 2 a b 2 a c 2 a b 1 2 b 2 2 b c 2 a c 2 b c 1 2 c 2 displaystyle begin pmatrix 1 2a 2 amp 2ab amp 2ac 2ab amp 1 2b 2 amp 2bc 2ac amp 2bc amp 1 2c 2 end pmatrix nbsp dd Takij pidhid pracyuye lishe yaksho ploshina prohodit cherez pochatok koordinat yaksho ni potribne afinne peretvorennya Ortogonalna proyekciyaDokladnishe Proekcijna matricyaDlya proyekcionuvannya vektora ortogonalno na pryamu yaka prohodit cherez pochatok koordinat poznachimo yak ux uy vektor sho lezhit na pryamij Todi matriceyu ortogonalnogo proektuvannya ye matricya A 1 u x 2 u y 2 u x 2 u x u y u x u y u y 2 displaystyle mathbf A frac 1 u x 2 u y 2 begin pmatrix u x 2 amp u x u y u x u y amp u y 2 end pmatrix nbsp dd Yak i z vidbittyam ortogonalna proyekciya na pryamu yaka ne prohodit cherez pochatok koordinat ye afinnim peretvorennyam a ne linijnim Vlastivosti RedaguvatiIzomorfizmi mizh prostorami linijnih vidobrazhen i matric Redaguvati Dlya vektornih prostoriv V displaystyle V nbsp i W displaystyle W nbsp yih bazisiv A a 1 a m displaystyle mathcal A mathbf a 1 ldots mathbf a m nbsp i B b 1 b n displaystyle mathcal B mathbf b 1 ldots mathbf b n nbsp i linijnogo vidobrazhennya T V W displaystyle T V to W nbsp matricya linijnogo vidobrazhennya M T A B displaystyle M T mathcal A mathcal B nbsp viznachena odnoznachno Navpaki dlya takih prostoriv i bazisiv kozhna m n displaystyle m times n nbsp matricya M a 11 a 1 j a 1 n a 21 a 2 j a 2 n a m 1 a m j a m n displaystyle M begin pmatrix a 11 amp dots amp a 1j amp dots amp a 1n a 21 amp dots amp a 2j amp dots amp a 2n vdots amp amp vdots amp amp vdots a m1 amp dots amp a mj amp dots amp a mn end pmatrix nbsp zadaye yedine linijne vidobrazhennya iz V displaystyle V nbsp u W displaystyle W nbsp Spravdi yaksho C c 1 c n displaystyle mathcal C mathbf c 1 cdots mathbf c n nbsp ye poslidovnistyu bud yakih vektoriv prostoru W displaystyle W nbsp ne obov yazkovo bazisom to isnuye yedine linijne vidobrazhennya T V W displaystyle T V rightarrow W nbsp z T a j c j displaystyle T mathbf a j mathbf c j nbsp dlya j 1 n displaystyle j 1 cdots n nbsp Cya yedina T displaystyle T nbsp viznachayetsya tak T x 1 a 1 x n a n T x 1 a 1 T x n a n x 1 T a 1 x n T a n x 1 c 1 x n c n displaystyle T x 1 mathbf a 1 cdots x n mathbf a n T x 1 mathbf a 1 cdots T x n mathbf a n x 1 T mathbf a 1 cdots x n T mathbf a n x 1 mathbf c 1 cdots x n mathbf c n nbsp Zvisno yaksho C c 1 c n displaystyle mathcal C mathbf c 1 cdots mathbf c n nbsp viyavitsya bazisom W displaystyle W nbsp todi T displaystyle T nbsp ce linijna biyekciya inakshe kazhuchi T displaystyle T nbsp ce izomorfizm Yaksho na dodatok do cogo W V displaystyle W V nbsp todi kazhut sho T displaystyle T nbsp ce avtomorfizm Takim chinom prostori linijnih vidobrazhen T V W displaystyle T V to W nbsp i matric rozmirnosti m n displaystyle m times n nbsp ye izomorfnimi vektornimi prostorami Prote izomorfizm mizh nimi zadanij tut zalezhit vid viboru bazisiv A displaystyle mathcal A nbsp i B displaystyle mathcal B nbsp Dlya inshogo viboru bazisiv oderzhuyetsya inshij izomorfizm tobto odnomu i tomu zh linijnomu vidobrazhennyu vidpovidatimut rizni matrici V matematichnij literaturi cherez ce inodi pishut sho izomorfizmi mizh linijnimi vidobrazhennyami i matricyami ne ye kanonichnimi Matricya kompoziciyi linijnih vidobrazhen Redaguvati Nehaj dodatkovo do poperednogo dano takozh vektornij prostir Z displaystyle Z nbsp rozmirnosti k displaystyle k nbsp nad tim zhe polem i linijne vidobrazhennya S W Z displaystyle S W to Z nbsp Nehaj na Z displaystyle Z nbsp vibrano bazis C c 1 c k displaystyle mathcal C mathbf c 1 cdots mathbf c k nbsp Todi analogichno do poperednogo mozhna viznachiti matricyu linijnogo vidobrazhennya M S B C displaystyle M S mathcal B mathcal C nbsp Dana matricya matime rozmirnist p m displaystyle p times m nbsp Kompoziciya vidobrazhen S T V Z displaystyle S circ T V to Z nbsp tobto vidobrazhennya dlya yakogo S T x S T x displaystyle S circ T x S T x nbsp bude linijnim vidobrazhennyam matriceyu yakogo bude dobutok matric M S B C displaystyle M S mathcal B mathcal C nbsp i M T A B displaystyle M T mathcal A mathcal B nbsp M S T A C M S B C M T A B displaystyle M S circ T mathcal A mathcal C M S mathcal B mathcal C cdot M T mathcal A mathcal B nbsp Zokrema yaksho T displaystyle T nbsp ye linijnim izomorfizmom to matriceyu obernenogo vidobrazhennya ye obernena matricya do matrici vidobrazhennya T displaystyle T nbsp M T 1 B A M T A B 1 displaystyle M T 1 mathcal B mathcal A M T mathcal A mathcal B 1 nbsp Zmina matrici pri perehodi do novih bazisiv Redaguvati Dokladnishe Zmina bazisuYak zaznacheno vishe matricya linijnogo vidobrazhennya zalezhit vid viboru bazisiv A displaystyle mathcal A nbsp i B displaystyle mathcal B nbsp vidpovidnih prostoriv Prote matrici dlya riznih bazisiv pov yazani prostoyu formuloyu iz vikoristannyam matric perehodu mizh bazisami Nehaj prostori V displaystyle V nbsp i W displaystyle W nbsp zadani yak i vishe dlya prostoru V displaystyle V nbsp vibrani dva bazisi A a 1 a n displaystyle mathcal A mathbf a 1 ldots mathbf a n nbsp i A a 1 a n displaystyle mathcal A mathbf a 1 ldots mathbf a n nbsp a u prostori W displaystyle W nbsp vibrani dva bazisi B b 1 b m displaystyle mathcal B mathbf b 1 ldots mathbf b m nbsp i B b 1 b m displaystyle mathcal B mathbf b 1 ldots mathbf b m nbsp Poznachimo B A A displaystyle B mathcal A mathcal A nbsp matricyu stovpci yakoyi ye koeficiyentami rozkladu vektoriv iz A displaystyle mathcal A nbsp cherez bazis A displaystyle mathcal A nbsp Ekvivalentno iz vikoristannyam mnozhennya vektor ryadka na matricyu dlya B A A displaystyle B mathcal A mathcal A nbsp vikonuyetsya spivvidnoshennya a 1 a n a 1 a n B A A displaystyle mathbf a 1 ldots mathbf a n mathbf a 1 ldots mathbf a n cdot displaystyle B mathcal A mathcal A nbsp de u pravij chastini vektori iz ryadka mnozhatsya na skalyari iz matrici Takozh yaksho deyakij vektor maye koordinati x 1 x n displaystyle begin pmatrix x 1 vdots x n end pmatrix nbsp u bazisi A displaystyle mathcal A nbsp i y 1 y n displaystyle begin pmatrix y 1 vdots y n end pmatrix nbsp u bazisi A displaystyle mathcal A nbsp to y 1 y n B A A x 1 x n displaystyle begin pmatrix y 1 vdots y n end pmatrix displaystyle B mathcal A mathcal A begin pmatrix x 1 vdots x n end pmatrix nbsp U cij formuli navpaki koordinati u bazisi A displaystyle mathcal A nbsp virazhayutsya cherez koordinati u bazisi A displaystyle mathcal A nbsp Matricya B A A displaystyle B mathcal A mathcal A nbsp nazivayetsya matriceyu perehodu vid bazisa A displaystyle mathcal A nbsp do bazisa A displaystyle mathcal A nbsp abo zgidno poperednogo u zvorotnomu poryadku vid koordinat u bazisi A displaystyle mathcal A nbsp do koordinat u bazisi A displaystyle mathcal A nbsp Analogichno mozhna viznachiti i matricyu B B B displaystyle B mathcal B mathcal B nbsp yaka nazivayetsya matriceyu perehodu vid bazisa B displaystyle mathcal B nbsp do bazisa B displaystyle mathcal B nbsp abo vid koordinat u bazisi B displaystyle mathcal B nbsp do koordinat u bazisi B displaystyle mathcal B nbsp Obidvi ci matrici ye nevirodzhenimi i B A A 1 B A A displaystyle B mathcal A mathcal A 1 B mathcal A mathcal A nbsp i B B B 1 B B B displaystyle B mathcal B mathcal B 1 B mathcal B mathcal B nbsp tobto oberneni matrici rivni matricyam zvorotnih perehodiv mizh bazisami Yaksho teper T V W displaystyle T V to W nbsp ye linijnim vidobrazhennyam i M T A B displaystyle M T mathcal A mathcal B nbsp i M T A B displaystyle M T mathcal A mathcal B nbsp ye jogo matricyami u riznih bazah to ci matrici zadovolnyayut spivvidnoshennya M T A B B B B M T A B B A A B B B 1 M T A B B A A displaystyle M T mathcal A mathcal B B mathcal B mathcal B cdot M T mathcal A mathcal B cdot B mathcal A mathcal A B mathcal B mathcal B 1 cdot M T mathcal A mathcal B cdot B mathcal A mathcal A nbsp Zokrema yaksho V W displaystyle V W nbsp i T displaystyle T nbsp ye linijnim peretvorennyam to jogo matrici u bazisah A displaystyle mathcal A nbsp i A displaystyle mathcal A nbsp pov yazani spivvidnoshennyam M T A B A A 1 M T A B A A displaystyle M T mathcal A B mathcal A mathcal A 1 cdot M T mathcal A cdot B mathcal A mathcal A nbsp U prostishih poznachennyah yaksho A displaystyle A nbsp ye matriceyu peretvorennya u bazisi A displaystyle mathcal A nbsp a A displaystyle A nbsp ye matriceyu peretvorennya u bazisi A displaystyle mathcal A nbsp i U B A A displaystyle U B mathcal A mathcal A nbsp to A U 1 A U displaystyle A U 1 AU nbsp Zauvazhennya U riznih avtoriv matriceyu perehodu vid bazisa A displaystyle mathcal A nbsp do bazisa A displaystyle mathcal A nbsp mozhe nazivatisya yak matricya B A A displaystyle B mathcal A mathcal A nbsp vvedena u cij statti tak i obernena do neyi matricya B A A displaystyle B mathcal A mathcal A nbsp yaka tut nazivayetsya matriceyu perehodu vid koordinat u bazisi A displaystyle mathcal A nbsp do koordinat u bazisi A displaystyle mathcal A nbsp Todi zokrema u ostannij formuli zamist matrici U displaystyle U nbsp vikoristovuyut yiyi obernenu V U 1 displaystyle V U 1 nbsp i formula podayetsya u inshomu poshirenomu vidi A V A V 1 displaystyle A VAV 1 nbsp Takozh pri vikoristanni zamist viznachenoyi u statti matrici vidobrazhennya yiyi transponovanoyi matrici peretvorennya tezh vikoristovuyut transponovani do viznachenih tut Cherez ci ta inshi prichini formuli sho pov yazuyut matrici vidobrazhen ta peretvoren pri zmini bazisiv popri svoyu prostotu ye prichinoyu chislennih pomilok inodi navit dosvidchenih matematikiv Matrici deyakih nelinijnih vidobrazhen RedaguvatiLinijne vidobrazhennya ne yedine yake mozhna predstaviti za dopomogoyu matric Deyaki peretvorennya sho ne ye linijnimi v evklidovomu prostori Rn mozhut buti predstavleni yak linijne peretvorennya u prostori rozmirnistyu n 1 Rn 1 V takomu vipadku vona vklyuchatime yak afinni peretvorennya taki yak peremishennya i proektivni peretvorennya Zokrema matrici peretvorennya 4 4 shiroko vikoristovuyutsya u zastosuvannyah trivimirnoyi komp yuternoyi grafiki Ci n 1 vimirni matrici peretvorennya nazivayutsya po riznomu v zalezhnosti vid oblasti yih zastosuvannya afinni matrici peretvorennya proektivni matrici peretvorennya abo v bilsh zagalnomu varianti matrici ne linijnogo peretvorennya Po vidnoshennyu do n vimirnoyi matrici matricya rozmirnistyu n 1 mozhe vvazhatisya rozshirenoyu matriceyu Dzherela RedaguvatiGelfand I M Lekcii po linejnoj algebre 5 e Moskva Nauka 1998 320 s ISBN 5791300158 ros Gantmaher F R Teoriya matric 5 e M Fizmatlit 2010 559 s ISBN 5 9221 0524 8 ros Lankaster P Teoriya matric Moskva Nauka 1973 280 s ros R Horn Ch Dzhonson Matrichnyj analiz M Mir 1989 653 s ros Weisstein Eric W Matricya obertannya angl na sajti Wolfram MathWorld Kalkulyator linijnih peretvoren Arhivovano 4 veresnya 2009 u Wayback Machine angl Aplet peretvoren Arhivovano 10 kvitnya 2010 u Wayback Machine angl Generuye matrici z 2D peretvoren i navpaki Primitki Redaguvati Gentle James E 2007 Matrix Transformations and Factorizations Matrix Algebra Theory Computations and Applications in Statistics Springer ISBN 9780387708737 Arhiv originalu za 21 lyutogo 2017 Procitovano 28 grudnya 2015 Otrimano z https uk wikipedia org w index php title Matricya linijnogo vidobrazhennya amp oldid 38078472
