Oriw Ro Jnow Mon, Sptmr 2, 24 si mtri oprtions (, -, *) Cross n ot prouts Dtrminnts n inrss Homonous oorints Ortonorml sis itionl Rsours 8.6 Tt ook 6.837 Tt ook 6.837-stff@rpis.sil.mit.u Ck t ours wsit for op of ts nots Wt is Mtri? mtri is st of lmnts, orni into rows n olumns m n mtri n olumns m rows si Oprtions si Oprtions Trnspos: Swp rows wit olumns M f i M T f i V V T [ ] ition n Sutrtion f f f f Just lmnts Just sutrt lmnts - - -
Multiplition si Oprtions Multiplition Is? M, ut m not! f f f Multipl row olumn f......... f f......... n m n n multipli n n p mtri to il n m p rsult Hs up: multiplition is NOT ommutti! Vtor Oprtions Vtor Intrprttion Vtor: n mtri Intrprttion: point or lin in n-imnsionl sp Dot Prout, Cross Prout, n Mnitu fin on tors onl r Tink of tor s lin in 2D or 3D Tink of mtri s trnsformtion on lin or st of lins ' ' V V Vtors: Dot Prout Vtors: Dot Prout Intrprttion: t ot prout msurs to wt r two tors r lin T f [ ] f Tink of t ot prout s mtri multiplition If n lnt θ os θ 2 T os(θ ) T mnitu is t ot prout of tor wit itslf T ot prout is lso rlt to t nl twn t two tors 2
Vtors: Cross Prout T ross prout of tors n is tor C wi is priulr to n T mnitu of C is proportionl to t sin of t nl twn n T irtion of C follows t rit n rul if w r workin in rit-n oorint sstm sin(θ ) Vtors: Cross Prout T ross-prout n omt s spill onstrut trminnt iˆ ˆj kˆ Inrs of Mtri Dtrminnt of Mtri Intit mtri: I Som mtris n inrs, su tt: - I Inrsion is trik: (C) - C - - - Dri from nonommuttiit proprt I Us for inrsion If t(), tn s no inrs Cn foun usin ftorils, piots, n oftors! Lots of intrprttions for mor info, tk 8.6 t( ) Dtrminnt of Mtri Inrs of Mtri f i f f i i f i f i f i For 3 3 mtri: Sum from lft to rit Sutrt from rit to lft Not: In t nrl s, t trminnt s n! trms f i. p t intit mtri to 2. Sutrt multipls of t otr rows from t first row to ru t ionl lmnt to 3. Trnsform t intit mtri s ou o 4. Wn t oriinl mtri is t intit, t intit s om t inrs! 3
Homonous Mtris Prolm: ow to inlu trnsltions in trnsformtions (n o prspti trnsforms) Solution: n tr imnsion ' ' ' 2 2 2 2 22 t t t Ortonorml sis sis: sp is totll fin st of tors n point is linr omintion of t sis Ortoonl: ot prout is ro Norml: mnitu is on Ortonorml: ortoonl norml Most ommon Empl: ˆ, ˆ, ˆ Cn of Ortonorml sis Cn of Ortonorml sis Gin: oorint frms n un point p (p, p, p) Fin: p (, p, ) n p u u p u. u. u.. u (. u) u (. u) u (. u) u n (. ) (. ) (. ) u (. n) n (. n) n (. n) n Cn of Ortonorml sis Cn of Ortonorml sis (. u) u (. u) u (. u) u (. ) (. ) (. ) (. n) n (. n) n (. n) n p (. u) u p [ (. u) u p [ (. u) u p [ (. ) (. ) (. ) (. n) n ] (. n) n ] (. n) n ] Sustitut into qution for p: p (p, p, p) p p p p p [ (. u) u p [ (. u) u p [ (. u) u (. ) (. ) (. ) (. n) n (. n) n (. n) n ] ] ] Rwrit: p [ p (. u) [ p (. ) [ p (. n) p (. u) p (. ) p (. n) p (. u) ] u p (. ) ] p (. n) ] n 4
Cn of Ortonorml sis Cn of Ortonorml sis p [ p (. u) [ p (. ) [ p (. n) p (. u) p (. ) p (. n) p (. u) ] u p (. ) ] p (. n) ] n p p (. u) p (. ) p (. n) p (. u) p (. ) p (. n) p (. u) p (. ) p (. n) p (, p, ) u p n Eprss in un sis: p p (. u) p (. ) p (. n) p (. u) p (. ) p (. n) p (. u) p (. ) p (. n) In mtri form: p u n u n u n p p p wr: u. u u. u t. Cn of Ortonorml sis Cts p Wt's M -, t inrs? p p p u u u p p n n n p u n u n p u n M p p p u. u u. u M - M T Rit-n s. lft-n oorint sstms OGL is rit-n Row-mjor s. olumn-mjor mtri stor. mtri. uss row-mjor orr OGL uss olumn-mjor orr 2 3 4 5 6 7 8 9 2 3 4 5 row-mjor 4 8 2 5 9 3 2 6 4 3 7 5 olumn-mjor Qustions?? 5