Inner Product Spaces INNER PRODUCTS

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MA4Hcdoc Ir Product Spcs INNER PRODCS Dto A r product o vctor spc V s ucto tht ssgs ubr spc V such wy tht th ollowg xos holds: P : w s rl ubr P : P : P 4 : P 5 : v, w = w, v v + w, u = u + w, u rv, w = r w v, v V w to vry pr w o vctor Dto A vctor spc V wth r product s clld r product spc Not, :V V R ( V, R,+, s vctor spc (, R, +,,, V s r product spc Expls h ollowg r r product spcs (, R, +,,, R, d X, Y = X Y th dot product ( C [, b], R, +,,,, d g = b ( x g( x (,,,,, A, B = tr AB R +, dx M, d ( hor Lt, b r product o spc V Lt u, w dot vctors V, r rl ubr h: u, v + w = u, v + u, w v, rw = r w v, = =, v 4 v, v = d oly v = hor I A s y postv dt trx, th X, Y = X AY, X, Y R ds r product o R, d vry r product o R, d vry r product o R rss ths wy Proo: Pg o 9

MA4Hcdoc X, Y = X AY s r product Ay, o R c b xprssd s X AY : = E b th stdrd bss o R h X = = Lt E { E,, } Morovr, Y = = X, Y y E = x E, y E = x y = = = E, E E, E E, E E, E [ x x ] = X AY A = A, = E, E y y x E d NORMS AND DISANCE Dto h or o v V s dd s or ( v v = v = (lgth Dto h dstc btw vctors w r product spc s ( w = v w d hor I v s y vctor r product spc, th ultpl o V v v ˆ = s th uqu ut vctor tht s postv v hor: Schwrz Iqulty I v d w r two vctors r product spc V, th d oly o o v or w s sclr ultpl o th othr w v w Morovr, qulty occurs Proo: Assu v = > d w = b > bv w = bv w, bv w = b( b w w b bv + w = bv + w, bv + w = b( b + w w b So b w b w b v w b =, d w Not: v w w or v w Pg o 9

MA4Hcdoc Expl Cosdr th vctor spc C [, b] o ll cotuous uctos o [, b] D g ( x g( x b b b h ( x g( x dx ( ( x dx ( g( x dx, dx = b hor: Proprts o Nors v v = oly v = rv = r v 4 v + w v + w (trgl qulty hor: Proprts o Dstc d(, w v d ( v, w = d oly v = w d ( v, w = d( w, v 4 d ( w d( u + d( u, w, u, w V ORHOGONAL SES OF VECORS Dto wo vctors w r product spc V r sd to b orthogol v, w = Dto A st o vctors {,, } ddto, s clld orthogol st ch d, =, I, =,, th th st s clld orthoorl st Expl = π π Cosdr { s x, cos x} C [ π,π ] wth g ( x g( x { x, cos x} s s orthogol st, dx h s x, cos x =, so hor: Pythgor hor s orthogol st o vctors, th + + = + + I {,, } hor Lt {,, } { r, } b orthogol st o vctors h:, r s lso orthogol or ll r R Pg o 9

MA4Hcdoc,, s orthoorl st,, s lrly dpdt { } hor: Expso hor Lt {,, } v = = b orthogol bss o r product spc V I v s y vctor V, th s th xpso o v s lr cobto o th bss vctors h cocts r clld Fourr cocts o v wth rspct to th orthogol bss {,, } L: Orthogol L Lt {,, } b orthogol st o vctors r product spc V, d lt v b y vctor ot sp{,, } D + = v =, th {, } s orthogol st o vctors,, + Gr-Schdt Orthogolzto Algorth Lt V b r product spc d { v,, } b y bss o V D vctors {,, } succssvly s ollows: = v v, = v v, v, = v v, = v = h {,, } s orthogol d { v,, v } sp{,, } v sp = V Dto h orthogol coplt o V s dd by = { v V u =, u } hor Lt b t dsol subspc o r product spc V h: s subspc o V d V = I d V =, th d + d = I d V =, th d = Proo o : s subspc o V bcus: Pg 4 o 9

MA4Hcdoc u + u V = bcus: Proo o : Sc Lt x, th x x, u =, u But x, x = sc x, so x = So = k bss { b,, } b d bss { b,, } b k + Assu sp{ b,, b, b+, bk } D v = v = b v b V, { b, b, b,, b, *}, k v k = * b So = k = + + s orthogol st V, so v *, = or =,, hs s V = + V = v* Cotrdcto! So V = sp{ b,, b, b+,, bk } d, so Proo o : = { v V ( u =, u } Dto ( v u d V = = d + d It s clr tht = pro : V V, pro = whr v = u + w or u, w W, V = W s clld th procto o wth krl W hor: Procto hor Lt b t dsol subspc o r product spc V d lt pro : V V s lr oprtor wth g d krl pro ( v d v ( v pro I {,, } s y orthogol bss o, th ( v = Proo o : pro : V V s lr oprtor bcus: v V pro = b h: Lt v = u + w d v = u + w pro ( v + v = pro ( u + u + w + w = u + u = pro ( v + pro ( v pro v = pro u + w = u = pro v ( ( ( ( pro = { pro ( v v V } k v u u ( pro = kr ( pro = { v V pro ( v = } pro ( v = pro ( u + u = u = kr ( = pro Proo o : pro ( v ollows ro dto Proo o : ( ( v pro v = u + u u = u =, h pro ( v = pro ( u u = So So Pg 5 o 9

MA4Hcdoc I {,, } s orthogol bss o, d {,, } th {,, +,, } s orthogol bss o V Sc, = v = +, v = = = + + s orthogol bss o = u + u, ( pro v = pro + = + = = = + = =, hor: Approxto hor Lt b t dsol subspc o r product spc V I tht s closst to v Closst s tht v pro ( v < v u, u, u pro ( v v V, th pro ( v s th vctor Expl Fd th polyol P tht bst pproxts th ucto ( x = x Assu [, ] = ( x g( x, g dx = {, x,x } B s orthogol bss o P x,x x, x, x pro ( x = + x + ( x = ( 5x + x P x 6 V d ORHOGONAL DIAGONALIZAION hor Lt : V V b lr oprtor o V h th ollowg codtos r quvlt: V hs bss o gvctors o M B s dgol hr xsts bss B o V such tht ( Proo: k B = {,, } bss o V h ( ( [ ( ( ] = λ M B B = λ λ hor Lt b lr oprtor o r product spc V I {,, } (, M ( = B Proo: Wrt ( [ ] M = B s orthogol bss o V, th Pg 6 o 9

MA4Hcdoc ( = + + C B ( ( Sc v = = = = or ll v V, ( = ( ( = = = = (, So Dto A lr oprtor s clld sytrc v ( w ( v, w, = holds or ll w V hor Lt V b t dsol r product spc h ollowg codtos r quvlt or lr oprtor : V V v, w = v, or ll w V ( ( w h trx o s sytrc wth rspct to vry orthoorl bss o V h trx o s sytrc wth rspct to so orthoorl bss o V 4 h s orthoorl bss {,, } o V such tht, ( (,, =, hor Lt : V V b sytrc lr oprtor o r product spc V, d lt b -vrt subspc o V h: h rstrcto o to s sytrc lr oprtor o s lso -vrt Proo: s tsl r product spc usg th s r product s V hus v, w = w, w, th, prtculr, t holds or w I ( ( V v d ( v u = ( v,, So v, u = hus u, th ( v u = ( u = u, u hor: Prcpl Axs hor h ollowg codtos r quvlt or lr oprtor o t dsol r product spc V s sytrc V hs orthogol bss cosstg o gvctor o Expl b gv by ( bx + cx = ( 8 b + c + ( + 5b + 4c x + ( + 4b + 5 c x Lt : P P + D + bx + cx, + b x + c x = + bb + cc Show tht sytrc d d orthoorl bss o P cosstg o gvctors Wt: s sytrc Pg 7 o 9

MA4Hcdoc k orthoorl bss o P {, x x } B =, 8 h M ( B = 5 4 So s sytrc 4 5 Wt: Orthoorl bss cosstg o gvctors W kow th gvlus o M ( d thus gvctors o ( {,, } =,, R, P such tht, W r lookg or { } I M B ( P M B ( P P P ( C ( C ( = s dgol, th [ ] [ ] B B = ( ( ( = So ( ( ( = + x x = + x + x = + x + x M to b ISOMERIES hor Lt : V V b lr oprtor o t dsol r product spc V h th ollowg codtos r quvlt: v = v, v ( prsrvs or ( V ( v ( v = v v v v V,, ( prsrvs dstc ( v, ( v = v, v V ( prsrvs r product 4 I {, } s y orthoorl bss V, th { (, ( } prsrvs bss Dto s lso orthoorl bss ( A lr oprtor s clld sotry t stss o o th codtos th prvous thor Corollry Evry sotry s soorphs h copost o two sotrs s sotry Expl Cosdr : d d A B tr( AB M M, = h ( A A = s sotry Pg 8 o 9

MA4Hcdoc hor Lt : V V b oprtor whr V s t dsol r product spc h th ollowg codtos r quvlt: s sotry M B s orthogol trx or vry orthoorl bss B ( M B ( s orthogol trx or so orthoorl bss B Proo: : Lt B = {,, } b orthoorl bss h th th colu o ( C Now C (, C =, sc C V R ( ( B d ( ( B ( ( ( ( ( B, =, sc : V V s sotry colus o M B ( r orthogol : Lt B {,, } = b th orthoorl bss h ( ( ( (, C ( ( { (,, ( } B M B s : s sotry,, =, =, so th,, =, B B = bcus M B ( s orthogol So, s orthoorl bss o V So s sotry Corollry I : V V s sotry whr V s t dsol r product spc, th dt = ± hor Lt : V V b sotry o two dsol r product spc V h thr r two possblts Ethr: cosθ s θ M B =, θ < (rotto s θ cosθ Or: M B =, θ < (rlcto hr s orthoorl bss B o V such tht ( π hr s orthoorl bss B o V such tht ( π L Lt : V V b sotry o t dsol r product spc V h: I s -vrt, th s lso -vrt I λ s coplx gvlus o, th λ = I hs o-rl gvlus, th V hs -dsol -vrt subspc Pg 9 o 9

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