U linijnij algebri vektor stovpec ce stovpec elementiv napriklad x x 1 x 2 x m displaystyle boldsymbol x begin bmatrix x 1 x 2 vdots x m end bmatrix Analogichno vektor ryadok ce ryadok elementiv 1 a a 1 a 2 a n displaystyle boldsymbol a begin bmatrix a 1 amp a 2 amp dots amp a n end bmatrix Vsyudi zhirnij kursivnij shrift vikoristovuyetsya yak dlya vektor ryadkiv tak i dlya vektor stovpciv Transponuvannya poznachayetsya yak T displaystyle rm T vektor ryadka ye vektor stovpcem x 1 x 2 x m T x 1 x 2 x m displaystyle begin bmatrix x 1 amp x 2 amp dots amp x m end bmatrix rm T begin bmatrix x 1 x 2 vdots x m end bmatrix a transponuvannya vektor stovpcya ye vektor ryadkom x 1 x 2 x m T x 1 x 2 x m displaystyle begin bmatrix x 1 x 2 vdots x m end bmatrix rm T begin bmatrix x 1 amp x 2 amp dots amp x m end bmatrix Sukupnist usih vektor ryadkiv z n displaystyle n elementami utvoryuye n displaystyle n vimirnij vektornij prostir analogichno mnozhina vsih vektor stovpciv z m displaystyle m elementami utvoryuye m displaystyle m vimirnij vektornij prostir Prostir vektor ryadkiv z n displaystyle n elementami mozhna rozglyadati yak dualnij prostir prostoru vektor stovpciv z n displaystyle n elementami oskilki bud yakij linijnij funkcional na prostori vektor stovpciv mozhna predstaviti yak mnozhennya zliva yedinogo vektor ryadka Zmist 1 Poznachennya 2 Operaciyi 3 Matrichni peretvorennya 4 Div takozh 5 Primitki 6 LiteraturaPoznachennya RedaguvatiShob sprostiti zapis vektor stovpciv u ryadku z inshim tekstom inodi yih zapisuyut yak vektor ryadki iz zastosuvannyam do nih operaciyi transponuvannya x x 1 x 2 x m T displaystyle boldsymbol x begin bmatrix x 1 amp x 2 amp dots amp x m end bmatrix rm T nbsp abo x x 1 x 2 x m T displaystyle boldsymbol x begin bmatrix x 1 x 2 dots x m end bmatrix rm T nbsp Deyaki avtori takozh vikoristovuyut domovlenist zapisu i vektor stovpciv i vektor ryadkiv yak ryadkiv ale rozdilyayuchi elementi vektor ryadka komami a elementi vektor stovpcya krapkami z komami div alternativne poznachennya 2 u tablici nizhche Vektor ryadok Vektor stovpecStandartne matrichne poznachennya probili v masivi bez kom znaki transponuvannya x 1 x 2 x m displaystyle begin bmatrix x 1 x 2 dots x m end bmatrix nbsp x 1 x 2 x m abo x 1 x 2 x m T displaystyle begin bmatrix x 1 x 2 vdots x m end bmatrix text abo begin bmatrix x 1 x 2 dots x m end bmatrix rm T nbsp Alternativne poznachennya 1 komi znaki transponuvannya x 1 x 2 x m displaystyle begin bmatrix x 1 x 2 dots x m end bmatrix nbsp x 1 x 2 x m T displaystyle begin bmatrix x 1 x 2 dots x m end bmatrix rm T nbsp Alternativne poznachennya 2 komi ta krapki z komami bez znakiv transponuvannya x 1 x 2 x m displaystyle begin bmatrix x 1 x 2 dots x m end bmatrix nbsp x 1 x 2 x m displaystyle begin bmatrix x 1 x 2 dots x m end bmatrix nbsp Operaciyi RedaguvatiMnozhennya matric vklyuchaye diyu mnozhennya kozhnogo vektor ryadka odniyeyi matrici na kozhen vektor stovpec inshoyi matrici Skalyarnij dobutok dvoh vektor stovpciv a displaystyle boldsymbol a nbsp i b displaystyle boldsymbol b nbsp ekvivalentnij matrichnomu dobutku transponovanogo vektora a displaystyle boldsymbol a nbsp ta vektora b displaystyle boldsymbol b nbsp a b a T b a 1 a n b 1 b n a 1 b 1 a n b n displaystyle boldsymbol a cdot boldsymbol b boldsymbol a rm T boldsymbol b begin bmatrix a 1 amp cdots amp a n end bmatrix begin bmatrix b 1 vdots b n end bmatrix a 1 b 1 cdots a n b n nbsp Vnaslidok simetriyi skalyarnogo dobutku dobutok dvoh vektor stovpciv a displaystyle boldsymbol a nbsp i b displaystyle boldsymbol b nbsp takozh ekvivalentnij matrichnomu dobutku transponovanogo vektora b displaystyle boldsymbol b nbsp ta vektora a displaystyle boldsymbol a nbsp b a b T a b 1 b n a 1 a n a 1 b 1 a n b n displaystyle boldsymbol b cdot boldsymbol a boldsymbol b rm T boldsymbol a begin bmatrix b 1 amp cdots amp b n end bmatrix begin bmatrix a 1 vdots a n end bmatrix a 1 b 1 cdots a n b n nbsp Matrichnij dobutok vektor stovpcya ta vektor ryadka daye vektornij dobutok dvoh vektoriv a displaystyle boldsymbol a nbsp i b displaystyle boldsymbol b nbsp yak priklad bilsh zagalnogo tenzornogo dobutku Matrichnij dobutok vektor stovpcya a displaystyle boldsymbol a nbsp ta vektor ryadka b displaystyle boldsymbol b nbsp daye elementi yihnogo diadichnogo dobutku a b a b T a 1 a 2 a 3 b 1 b 2 b 3 a 1 b 1 a 1 b 2 a 1 b 3 a 2 b 1 a 2 b 2 a 2 b 3 a 3 b 1 a 3 b 2 a 3 b 3 displaystyle boldsymbol a otimes boldsymbol b boldsymbol a boldsymbol b rm T begin bmatrix a 1 a 2 a 3 end bmatrix begin bmatrix b 1 amp b 2 amp b 3 end bmatrix begin bmatrix a 1 b 1 amp a 1 b 2 amp a 1 b 3 a 2 b 1 amp a 2 b 2 amp a 2 b 3 a 3 b 1 amp a 3 b 2 amp a 3 b 3 end bmatrix nbsp yakij ye transponuvannyam matrichnogo dobutku vektor stovpcya b displaystyle boldsymbol b nbsp i vektor ryadka a displaystyle boldsymbol a nbsp b a b a T b 1 b 2 b 3 a 1 a 2 a 3 b 1 a 1 b 1 a 2 b 1 a 3 b 2 a 1 b 2 a 2 b 2 a 3 b 3 a 1 b 3 a 2 b 3 a 3 displaystyle boldsymbol b otimes boldsymbol a boldsymbol b boldsymbol a rm T begin bmatrix b 1 b 2 b 3 end bmatrix begin bmatrix a 1 amp a 2 amp a 3 end bmatrix begin bmatrix b 1 a 1 amp b 1 a 2 amp b 1 a 3 b 2 a 1 amp b 2 a 2 amp b 2 a 3 b 3 a 1 amp b 3 a 2 amp b 3 a 3 end bmatrix nbsp Matrichni peretvorennya RedaguvatiDokladnishe Matricya linijnogo vidobrazhennyan n displaystyle n times n nbsp matricyu M displaystyle M nbsp mozhna predstaviti yak linijne vidobrazhennya ta diyati na vektor ryadki ta vektor stovpci yak matricya peretvorennya linijnogo vidobrazhennya Dlya vektor ryadka v displaystyle boldsymbol v nbsp dobutok v M displaystyle boldsymbol v M nbsp ye inshim vektor ryadkom p displaystyle boldsymbol p nbsp v M p displaystyle boldsymbol v M boldsymbol p nbsp Insha n n displaystyle n times n nbsp matricya Q displaystyle Q nbsp mozhe diyati na p displaystyle boldsymbol p nbsp p Q t displaystyle boldsymbol p Q boldsymbol t nbsp Todi mozhna zapisati t p Q v M Q displaystyle boldsymbol t boldsymbol p Q boldsymbol v MQ nbsp Otzhe peretvorennya matrichnogo dobutku M Q displaystyle MQ nbsp vidobrazhaye v displaystyle boldsymbol v nbsp bezposeredno v t displaystyle boldsymbol t nbsp Prodovzhuyuchi robotu z vektor ryadkami matrichni peretvorennya yaki dodatkovo perekonfiguruyut n displaystyle n nbsp prostir mozhna zastosuvati sprava na vihidni dani Yaksho vektor stovpec peretvoryuyetsya v inshij vektor stovpec pid diyeyu n n displaystyle n times n nbsp matrici to operaciya vidbuvayetsya zliva p T M v T t T Q p T displaystyle boldsymbol p rm T M boldsymbol v rm T quad boldsymbol t rm T Q boldsymbol p rm T nbsp sho privodit do algebrayichnogo spivvidnoshennya Q M v T displaystyle QM boldsymbol v rm T nbsp dlya skomponovanih vihidnih danih yaki otrimano z vhidnih danih v T displaystyle boldsymbol v rm T nbsp Matrichni peretvorennya roztashovuyutsya zliva pri takomu vikoristanni vektor stovpcya dlya vhodu v matrichne peretvorennya Div takozh RedaguvatiKovariantnist i kontravariantnist matematika Indeksne poznachennya en Matricya odinic Matrichna odinicya en Standartnij bazis Odinichnij vektorPrimitki Redaguvati Meyer 2000 p 8Literatura RedaguvatiAxler Sheldon Jay 1997 Linear Algebra Done Right 2nd ed Springer Verlag ISBN 0 387 98259 0 Lay David C August 22 2005 Linear Algebra and Its Applications 3rd ed Addison Wesley ISBN 978 0 321 28713 7 Meyer Carl D February 15 2001 Matrix Analysis and Applied Linear Algebra Society for Industrial and Applied Mathematics SIAM ISBN 978 0 89871 454 8 archived from the original on March 1 2001 Poole David 2006 Linear Algebra A Modern Introduction 2nd ed Brooks Cole ISBN 0 534 99845 3 Anton Howard 2005 Elementary Linear Algebra Applications Version 9th ed Wiley International Leon Steven J 2006 Linear Algebra With Applications 7th ed Pearson Prentice Hall Otrimano z https uk wikipedia org w index php title Vektor ryadki ta vektor stovpci amp oldid 36166790
