U linijnij algebri bazis dlya vektornogo prostoru ce linijno nezalezhna mnozhina dlya yakoyi cej prostir ye linijnoyu obolonkoyu 1 2 3 Cya stattya zdebilshogo rozglyadaye skinchenno vimirni vektorni prostori ale bagato teorem mayut misce dlya neskinchenno vimirnih vektornih prostoriv 3 Bazis vektornogo prostoru rozmirnosti n ce mnozhina z n vektoriv a1 an yaki nazivayut bazisnimi vektorami z vlastivistyu sho bud yakij vektor cogo prostoru mozhna predstaviti yak unikalnu linijnu kombinaciyu bazisnih vektoriv 4 5 3 Matrici perehodu operatoriv takozh viznacheni vibranim bazisom Cherez te sho chasto bazhano pracyuvati z bilshe nizh odnim bazisom dlya vektornogo prostoru u linijnij algebri zasadnicho vazhlivo buti zdatnim legko perehoditi vid koordinatnih predstavlen vektoriv i operatoriv v odnomu bazisi do yih totozhnih predstavlen v inshomu bazisi Takij perehid nazivayetsya zminoyu bazisu 6 7 8 Linijni kombinaciyi odniyeyi bazisnoyi mnozhini vektoriv fioletovi formuyut novi vektori chervoni Yaksho voni linijno nezalezhni to voni utvoryuyut novu bazisnu mnozhinu Linijni kombinaciyi sho pov yazuyut pershu mnozhinu i drugu stanovlyat linijne vidobrazhennya yake nazivayetsya zminoyu bazisu Vektor predstavleno v dvoh riznih bazisah fioletovi i chervoni strilki Hocha simvol R sho mi jogo vikoristovuyemo nizhche mozhe poznachati pole dijsnih chisel rezultati dijsni i yaksho R zamineno na bud yake pole F Hocha nizhche vikoristano terminologiyu vektornih prostoriv obgovoreni rezultati dijsni i todi koli R ce komutativne kilce a vektornij prostir povsyudno zamineno na vilnij R modul Zmist 1 Matricya perehodu 1 1 Oznachennya 1 2 Vlastivosti 2 Peretvorennya koordinat vektora pri zmini bazisu 2 1 Prikladi 2 1 1 Dva vimiri 2 1 2 Tri vimiri 3 Peretvorennya matrici linijnogo vidobrazhennya pri zmini bazisu 4 Matricya bilinijnoyi formi 4 1 Zmina bazisu 5 Dzherela 6 PrimitkiMatricya perehodu RedaguvatiOznachennya Redaguvati Matriceyu perehodu v n displaystyle n nbsp vimirnomu prostori vid bazisu A a 1 a 2 a n displaystyle mathcal A langle mathbf a 1 mathbf a 2 ldots mathbf a n rangle nbsp do bazisu B b 1 b 2 b n displaystyle mathcal B langle mathbf b 1 mathbf b 2 ldots mathbf b n rangle nbsp nazivayetsya kvadratna matricya stovpci yakoyi koordinati rozkladu vektoriv b 1 b 2 b n displaystyle langle mathbf b 1 mathbf b 2 ldots mathbf b n rangle nbsp u bazisi a 1 a 2 a n displaystyle langle mathbf a 1 mathbf a 2 ldots mathbf a n rangle nbsp A same nehaj vikonuyutsya rivnosti de vsi koeficiyenti odnoznachno viznacheni bo a 1 a 2 a n displaystyle langle mathbf a 1 mathbf a 2 ldots mathbf a n rangle nbsp ye bazisom b 1 a 11 a 1 a 12 a 2 a 1 n a n displaystyle mathbf b 1 alpha 11 mathbf a 1 alpha 12 mathbf a 2 ldots alpha 1n mathbf a n nbsp b 2 a 21 a 1 a 22 a 2 a 2 n a n displaystyle mathbf b 2 alpha 21 mathbf a 1 alpha 22 mathbf a 2 ldots alpha 2n mathbf a n nbsp displaystyle cdots nbsp b n a n 1 a 1 a n 2 a 2 a n n a n displaystyle mathbf b n alpha n1 mathbf a 1 alpha n2 mathbf a 2 ldots alpha nn mathbf a n nbsp Todi matricya perehodu maye viglyad B A B a 11 a 21 a n 1 a 12 a 22 a n 2 a 1 n a 2 n a n n displaystyle B mathcal A mathcal B begin pmatrix alpha 11 amp alpha 21 amp ldots amp alpha n1 alpha 12 amp alpha 22 amp ldots amp alpha n2 vdots amp vdots amp ddots amp vdots alpha 1n amp alpha 2n amp ldots amp alpha nn end pmatrix nbsp Yaksho zapisuvati bazisi za dopomogoyu vektor ryadkiv elementami yakih ye bazisni vektori to mozhna u matrichnij formi zapisati b 1 b 2 b n a 1 a 2 a n B A B a 1 a 2 a n a 11 a 21 a n 1 a 12 a 22 a n 2 a 1 n a 2 n a n n displaystyle langle mathbf b 1 mathbf b 2 ldots mathbf b n rangle langle mathbf a 1 mathbf a 2 ldots mathbf a n rangle cdot B mathcal A mathcal B langle mathbf a 1 mathbf a 2 ldots mathbf a n rangle begin pmatrix alpha 11 amp alpha 21 amp ldots amp alpha n1 alpha 12 amp alpha 22 amp ldots amp alpha n2 vdots amp vdots amp ddots amp vdots alpha 1n amp alpha 2n amp ldots amp alpha nn end pmatrix nbsp Vlastivosti Redaguvati Matriceyu perehodu vid dovilnogo bazisu A a 1 a 2 a n displaystyle mathcal A langle mathbf a 1 mathbf a 2 ldots mathbf a n rangle nbsp do samogo sebe ye odinichna matricya Yaksho A a 1 a 2 a n displaystyle mathcal A langle mathbf a 1 mathbf a 2 ldots mathbf a n rangle nbsp B b 1 b 2 b n displaystyle mathcal B langle mathbf b 1 mathbf b 2 ldots mathbf b n rangle nbsp i C c 1 c 2 c n displaystyle mathcal C langle mathbf c 1 mathbf c 2 ldots mathbf c n rangle nbsp ye troma bazisami odnogo vektornogo prostoru i B A B displaystyle B mathcal A mathcal B nbsp ye matriceyu perehodu vid A displaystyle mathcal A nbsp do bazisu B displaystyle mathcal B nbsp a B B C displaystyle B mathcal B mathcal C nbsp ye matriceyu perehodu vid bazisu B displaystyle mathcal B nbsp do bazisu C displaystyle mathcal C nbsp to matricya perehodu vid A displaystyle mathcal A nbsp do C displaystyle mathcal C nbsp ye dobutkom cih matric B A C B B C B A B displaystyle B mathcal A mathcal C B mathcal B mathcal C cdot B mathcal A mathcal B nbsp dd Zokrema iz poperednogo viplivaye sho matricya perehodu mizh bud yakimi matricyami ye nevirodzhenoyu i matricya zvorotnogo perehodu ye obernenoyu do danoyi matrici perehodu B B A B A B 1 displaystyle B mathcal B mathcal A B mathcal A mathcal B 1 nbsp dd Yaksho rozglyadayetsya vektornij prostir nad polem dijsnih chisel i bazis A a 1 a 2 a n displaystyle mathcal A langle mathbf a 1 mathbf a 2 ldots mathbf a n rangle nbsp ye ortonormovanim shodo deyakogo skalyarnogo dobutku na prostori to bazis B b 1 b 2 b n displaystyle mathcal B langle mathbf b 1 mathbf b 2 ldots mathbf b n rangle nbsp bude ortonormovanim todi i tilki todi koli matricya perehodu B A B displaystyle B mathcal A mathcal B nbsp bude ortogonalnoyu U vipadku kompleksnih vektornih prostoriv take same tverdzhennya spravedlive dlya unitarnih matric i ermitovih skalyarnih dobutkiv Peretvorennya koordinat vektora pri zmini bazisu RedaguvatiNehaj deyakij dovilnij vektor x displaystyle mathbf x nbsp virazhayetsya cherez vektori u bazisah A a 1 a 2 a n displaystyle mathcal A langle mathbf a 1 mathbf a 2 ldots mathbf a n rangle nbsp i B b 1 b 2 b n displaystyle mathcal B langle mathbf b 1 mathbf b 2 ldots mathbf b n rangle nbsp yak x x 1 a 1 x 2 a 2 x n a n i x i a i displaystyle mathbf x x 1 mathbf a 1 x 2 mathbf a 2 dots x n mathbf a n sum i x i mathbf a i nbsp i x y 1 b 1 y 2 b 2 y n b n i y i b i displaystyle mathbf x y 1 mathbf b 1 y 2 mathbf b 2 dots y n mathbf b n sum i y i mathbf b i nbsp Ci rivnosti dozvolyayut vvesti koordinatni vektor stovpci i za dopomogoyu matrichnogo dobutku i oznachennya matrici perehodu zapisati x a 1 a 2 a n x 1 x n b 1 b 2 b n y 1 y n a 1 a 2 a n B A B y 1 y n displaystyle mathbf x langle mathbf a 1 mathbf a 2 ldots mathbf a n rangle begin pmatrix x 1 vdots x n end pmatrix langle mathbf b 1 mathbf b 2 ldots mathbf b n rangle begin pmatrix y 1 vdots y n end pmatrix langle mathbf a 1 mathbf a 2 ldots mathbf a n rangle cdot B mathcal A mathcal B begin pmatrix y 1 vdots y n end pmatrix nbsp Iz odnoznachnosti zapisu vektora cherez bazis zvidsi viplivaye formula peretvorennya koordinat pri zmini bazisu x 1 x n B A B y 1 y n displaystyle begin pmatrix x 1 vdots x n end pmatrix B mathcal A mathcal B begin pmatrix y 1 vdots y n end pmatrix nbsp Tobto yaksho koordinati deyakogo vektora u bazisi B displaystyle mathcal B nbsp utvoryuyut vektor stovpec y displaystyle y nbsp a u bazisi A displaystyle mathcal A nbsp utvoryuyut vektor stovpec x displaystyle x nbsp to x B A B y displaystyle x B mathcal A mathcal B y nbsp Vazhlivo pomititi zminu poryadku u cij formuli Yaksho matricya B A B displaystyle B mathcal A mathcal B nbsp viznachaye perehid vid bazisu A displaystyle mathcal A nbsp do bazisu B displaystyle mathcal B nbsp to formula peretvorennya koordinat zadaye perehid navpaki vid koordinat u bazisi B displaystyle mathcal B nbsp do koordinat u bazisi A displaystyle mathcal A nbsp Tomu matricyu B A B displaystyle B mathcal A mathcal B nbsp mozhna takozh nazivati matriceyu perehodu vid koordinat u bazisi B displaystyle mathcal B nbsp do koordinat u bazisi A displaystyle mathcal A nbsp U takij interpretaciyi mozhna takozh dati oznachennya matrici perehodu cherez matrici linijnogo vidobrazhennya Stovpcyami takoyi matrici M T A B displaystyle M T mathcal A mathcal B nbsp ye koordinati T a i displaystyle T mathbf a i nbsp u bazisi B displaystyle mathcal B nbsp Yaksho vibrati totozhne linijne peretvorennya to stovpcyami matrici M I B A displaystyle M I mathcal B mathcal A nbsp budut koordinati rozkladiv vektoriv iz B displaystyle mathcal B nbsp u bazisi A displaystyle mathcal A nbsp Tomu B A B M I B A displaystyle B mathcal A mathcal B M I mathcal B mathcal A nbsp Zmina poryadku bazisiv u pravij i livij chastini ne ye pomilkovo Prikladi Redaguvati Dva vimiri Redaguvati U dvovimirnomu prostori dvijka vektoriv otrimanih obertannyam standartnogo bazisu proti godinnikovoyi strilki na 45 ye bazisom prostoru Matricya chiyi stovpchiki ye koordinatami cih vektoriv u pochatkovomu bazisi maye vid M 1 2 1 2 1 2 1 2 displaystyle M begin bmatrix frac 1 sqrt 2 amp frac 1 sqrt 2 frac 1 sqrt 2 amp frac 1 sqrt 2 end bmatrix nbsp Yaksho mi hochemo perevesti bud yakij vektor prostoru v cej bazis nam treba pomnozhiti zliva jogo komponenti na obernenu do ciyeyi matricyu 9 a shob perevesti vektor z koordinatami u novomu bazisi u koordinati standartnogo potribno novi koordinati pomnozhiti na samu matricyu Tri vimiri Redaguvati Nehaj R bude novim bazisom zadanim za dopomogoyu kutiv Ejlera Matricya cogo bazisu v yakosti stovpciv matime komponenti kozhnogo z vektoriv u standartnomu bazisi Otzhe cya matricya viglyadaye tak Div stattyu Ejlerovi kuti R c a c g s a c b s g c a s g s a c b c g s b s a s a c g c a c b s g s a s g c a c b c g s b c a s b s g s b c g c b displaystyle mathbf R begin bmatrix mathrm c alpha mathrm c gamma mathrm s alpha mathrm c beta mathrm s gamma amp mathrm c alpha mathrm s gamma mathrm s alpha mathrm c beta mathrm c gamma amp mathrm s beta mathrm s alpha mathrm s alpha mathrm c gamma mathrm c alpha mathrm c beta mathrm s gamma amp mathrm s alpha mathrm s gamma mathrm c alpha mathrm c beta mathrm c gamma amp mathrm s beta mathrm c alpha mathrm s beta mathrm s gamma amp mathrm s beta mathrm c gamma amp mathrm c beta end bmatrix nbsp Znov taki bud yakij vektor prostoru mozhna perevesti v cej novij bazis domnozhuyuchi jogo zliva na obernenu do ciyeyi matrici Peretvorennya matrici linijnogo vidobrazhennya pri zmini bazisu RedaguvatiNehaj zadani vektorni prostori V displaystyle V nbsp i W displaystyle W nbsp nad odnim polem i 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 Nehaj B A A displaystyle B mathcal A mathcal A nbsp i B B B displaystyle B mathcal B mathcal B nbsp ye vidpovidnimi perehodami mizh bazisami u dvoh prostorah Yaksho teper T V W displaystyle T V to W nbsp ye linijnim vidobrazhennyam to u vidpovidnih bazisah vono zadayetsya matricyami 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 Yaksho x V displaystyle mathbf x in V nbsp ye dovilnim vektorom koordinati yakogo u bazisah A displaystyle mathcal A nbsp i A displaystyle mathcal A nbsp mozhna zapisati za dopomogoyu vektor stovpciv x 1 x n displaystyle begin pmatrix x 1 vdots x n end pmatrix nbsp i x 1 x n displaystyle begin pmatrix x 1 vdots x n end pmatrix nbsp to T x displaystyle T mathbf x nbsp ye vektorom prostoru W displaystyle W nbsp koordinati yakogo u bazisah B B displaystyle mathcal B mathcal B nbsp mozhna zapisati za dopomogoyu vektor stovpciv y 1 y m displaystyle begin pmatrix y 1 vdots y m end pmatrix nbsp i y 1 y m displaystyle begin pmatrix y 1 vdots y m end pmatrix nbsp U cih poznachennyah u matrichnomu zapisi vrahovuyuchi oznachennya matric perehodu i linijnogo vidobrazhennya y 1 y m B B B y 1 y m B B B M T A B x 1 x n B B B M T A B B A A x 1 x n displaystyle begin pmatrix y 1 vdots y m end pmatrix B mathcal B mathcal B cdot begin pmatrix y 1 vdots y m end pmatrix B mathcal B mathcal B cdot M T mathcal A mathcal B cdot begin pmatrix x 1 vdots x n end pmatrix B mathcal B mathcal B cdot M T mathcal A mathcal B cdot B mathcal A mathcal A cdot begin pmatrix x 1 vdots x n end pmatrix nbsp Oskilki vkazani rivnosti spravedlivi dlya koordinatnih stovpciv usih vektoriv x V displaystyle mathbf x in V nbsp to B B B M T A B B A A displaystyle B mathcal B mathcal B cdot M T mathcal A mathcal B cdot B mathcal A mathcal A nbsp ye odnoznachno viznachenoyu matriceyu vidobrazhennya T V W displaystyle T V to W nbsp u bazisah A displaystyle mathcal A nbsp i B displaystyle mathcal B nbsp 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 Matricya bilinijnoyi formi RedaguvatiBilinijna forma na vektornomu prostori V nad polem R ce vidobrazhennya V V R linijne shodo oboh argumentiv Tobto B V V R bilinijna yaksho vidobrazhennya x B x y displaystyle mathbf x mapsto B mathbf x mathbf y nbsp y B y x displaystyle mathbf y mapsto B mathbf y mathbf x nbsp linijni dlya bud yakogo y z V Ce viznachennya takozh zastosovne dlya modulya nad komutativnim kilcem i gomomorfizmom modulya v yakosti linijnogo vidobrazhennya Matricya Grama G sho vidpovidaye bazisu A a 1 a 2 a n displaystyle mathcal A langle mathbf a 1 mathbf a 2 ldots mathbf a n rangle nbsp viznachena tak G i j B a i a j displaystyle G i j B mathbf a i mathbf a j nbsp Yaksho x i x i a i displaystyle mathbf x sum i x i mathbf a i nbsp i y i y i a i displaystyle mathbf y sum i y i mathbf a i nbsp ce predstavlennya vektoriv x y u comu bazisi todi bilinijna forma zadana tak B x y x T G y displaystyle B mathbf x mathbf y mathbf x mathsf T G mathbf y nbsp Matricya bude simetrichna yaksho bilinijna forma B ce simetrichna bilinijna forma Zmina bazisu Redaguvati Yaksho zadano dva bazisi A a 1 a 2 a n displaystyle mathcal A langle mathbf a 1 mathbf a 2 ldots mathbf a n rangle nbsp i A a 1 a 2 a n displaystyle mathcal A langle mathbf a 1 mathbf a 2 ldots mathbf a n rangle nbsp G displaystyle G nbsp ye matriceyu Grama u pershomu bazisi a G displaystyle G nbsp ye matriceyu grama u drugomu bazisi to ci matrici pov yazana spivvidnoshennyam iz matriceyu perehodu B A A displaystyle B mathcal A mathcal A nbsp G B A A T G B A A displaystyle G B mathcal A mathcal A mathsf T cdot G cdot B mathcal A mathcal A nbsp Dzherela RedaguvatiGelfand I M Lekcii po linejnoj algebre 5 e Moskva Nauka 1998 320 s ISBN 5791300158 ros Anton Howard 1987 Elementary Linear Algebra vid 5th New York Wiley ISBN 0 471 84819 0 Beauregard Raymond A Fraleigh John B 1973 A First Course In Linear Algebra with Optional Introduction to Groups Rings and Fields Boston Houghton Mifflin Company ISBN 0 395 14017 X Nering Evar D 1970 Linear Algebra and Matrix Theory vid 2nd New York Wiley LCCN 76091646 Primitki Redaguvati Anton 1987 s 171 Beauregard ta Fraleigh 1973 s 93 a b v Nering 1970 s 15 Anton 1987 s 74 76 Beauregard ta Fraleigh 1973 s 194 195 Anton 1987 s 221 237 Beauregard ta Fraleigh 1973 s 240 243 Nering 1970 s 50 52 Change of Basis HMC Calculus Tutorial www math hmc edu Arhiv originalu za 16 lipnya 2016 Procitovano 22 serpnya 2017 i poyasnennya dovedennya Why www math hmc edu Arhiv originalu za 22 serpnya 2017 Procitovano 22 serpnya 2017 Otrimano z https uk wikipedia org w index php title Zmina bazisu amp oldid 39346131
