Orthogonal vs Orthonormal
Hauv kev ua lej, ob lo lus orthogonal thiab orthonormal feem ntau siv nrog rau cov txheej txheem vectors. Ntawm no, lo lus 'vector' yog siv nyob rau hauv qhov kev nkag siab tias nws yog ib qho chaw ntawm vector chaw - ib qho qauv algebraic siv nyob rau hauv linear algebra. Rau peb qhov kev sib tham, peb yuav xav txog qhov chaw khoom sab hauv - qhov chaw vector V nrog rau cov khoom sab hauv txhais ntawm V.
Raws li piv txwv, rau cov khoom sab hauv, qhov chaw yog qhov teeb tsa ntawm txhua qhov 3-dimensional txoj hauj lwm vectors nrog rau cov khoom dot ib txwm.
Dab tsi yog orthogonal?
A nonnempty subset S ntawm qhov chaw khoom sab hauv V tau hais tias yog orthogonal, yog thiab tsuas yog rau txhua qhov sib txawv u, v hauv S, [u, v]=0; i.e. cov khoom sab hauv ntawm u thiab v yog sib npaug rau xoom scalar nyob rau hauv qhov khoom sab hauv.
Piv txwv li, nyob rau hauv lub teeb ntawm tag nrho 3-dimensional txoj hauj lwm vectors, qhov no yog sib npaug hais tias, rau txhua tus khub ntawm txoj hauj lwm vectors p thiab q hauv S, p thiab q yog perpendicular rau ib leeg. (Nco ntsoov tias cov khoom sab hauv hauv qhov chaw vector no yog cov khoom dot. Tsis tas li ntawd, cov khoom dot ntawm ob vectors yog sib npaug rau 0 yog thiab tsuas yog tias ob vectors yog perpendicular rau ib leeg.)
Xav txog qhov teeb tsa S={(0, 2, 0), (4, 0, 0), (0, 0, 5)}, uas yog ib feem ntawm 3-dimensional txoj hauj lwm vectors. Saib (0, 2, 0).(4, 0, 0)=0, (4, 0, 0).(0, 0, 5)=0 & (0, 2, 0).(0, 0)., 5)=0. Li no, lub teeb S yog orthogonal. Tshwj xeeb, ob vectors tau hais tias yog orthogonal yog tias lawv cov khoom sab hauv yog 0. Yog li ntawd, txhua khub vectors hauv Sis orthogonal.
Dab tsi yog orthonormal?
A tsis muaj qhov tsis sib xws S ntawm qhov chaw khoom sab hauv V tau hais tias yog orthonormal yog thiab tsuas yog S yog orthogonal thiab rau txhua vector u hauv S, [u, u]=1. Yog li ntawd, nws tuaj yeem pom tau tias txhua lub teeb orthonormal yog orthogonal tab sis tsis vice versa.
Piv txwv li, nyob rau hauv cov txheej txheem ntawm tag nrho 3-dimensional txoj hauj lwm vectors, qhov no yog sib npaug li hais tias, rau txhua tus khub ntawm txoj hauj lwm vectors p thiab q hauv S, p thiab q yog perpendicular rau ib leeg, thiab rau txhua p hauv S, |p|=1. Qhov no yog vim tus mob [p, p]=1 txo rau p.p=|p||p|p|cos0=|p|2=1, uas yog sib npaug rau |p |=1. Yog li ntawd, muab ib qho orthogonal teeb peb ib txwm tuaj yeem tsim ib qho kev sib thooj orthonormal los ntawm kev faib txhua vector los ntawm nws qhov loj.
T={(0, 1, 0), (1, 0, 0), (0, 0, 1)} yog ib qho orthonormal subset ntawm cov teeb ntawm tag nrho 3-dimensional txoj hauj lwm vectors. Nws yog qhov yooj yim kom pom tias nws tau los ntawm kev faib txhua qhov vectors hauv qhov teeb tsa S, los ntawm lawv qhov loj.
Qhov txawv ntawm orthogonal thiab orthonormal yog dab tsi?
- Ib qhov tsis muaj npe S ntawm qhov chaw khoom sab hauv V tau hais tias yog orthogonal, yog tias thiab tsuas yog rau txhua qhov sib txawv u, v hauv S, [u, v]=0. Txawm li cas los xij, nws yog orthonormal, yog tias thiab tsuas yog ib qho xwm txheej ntxiv - rau txhua tus vector u hauv S, [u, u]=1 txaus siab.
- Txhua qhov teeb tsa orthonormal yog orthogonal tab sis tsis hloov pauv.
- Txhua qhov teeb tsa orthogonal sib haum rau cov teeb tsa tshwj xeeb tab sis cov txheej txheem orthonormal tuaj yeem sib haum rau ntau qhov teeb tsa orthogonal.
