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Lecture20 - Led—«ME 20'TharsAay 00 23> 100 W 0 We“...

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Unformatted text preview: Led—«ME 20, 'TharsAay 00* 23> , 100% W_ 0 We“ 53% 5% Recall ”wed iRn ‘5 H36" 58" 0P (1“ 1’1 x l Co lama vecmg. Def—1n i " SubSPCkies 0F iRn “ LeF U 'be a bubsef of IR“ (Le. U 9. R"). «7mm U m gamed a “55533 a?» R" .[— .+ SahSFlF/o cw. 012 H11 'féllomna Psopuhes (SI) The laro ved‘or lb m U (.e EGU) ( Le. U ;s ClogseA under AAA-+100) (133) If: X is m U MA a ks a reai number +11%) ax 3s m U (Le. U 2‘s dosecl MA,“ Sad“, mdhpmfibn ) Expi : \) i?“ Sa‘l’bFIPS (5%) I‘ (52-) 04H (53), Hengp IS a $obsPaCe 9P ”2 I Law/66 '1; 6? (32.) and (33): Nofe ‘Hxaf fi‘le Plume. M is “he sgr i We l? Such Wad < x}, n > =0} , where {12)}? J «2 LJ {9: and E: be “’30 demefltS of? H' Thfl‘f iS/ <1§,5’>r—-0 and <§fl7>30 Wm fix, "3 <3. , a; + <5 z Henu {9: a- 3L bdofljb to H ._., ..., ‘7" “-7 ‘ . Hoveover, <00). «2 I" > “'1 a < ‘7' 2 n > 7’ (1(0) 730 » Hence 01—9: belongs to H ., Therefore H Sci’risfies ($2) and (5:3) , kenm H is ‘ , 3 o SU‘DSPQUI 44 n2 . f2] . , V , 2. . , Expz 2 The Ear P: f [g] C— “? Suck “mt Dig-o} . Z :5 not a wbgpace, Gr— 1R Prom): .} ['7] Efiongs f0 P Bid, ("4)[33] 509$ {70+ \Oebha to P Hens? P A090 no+ SAMSF/ (53); Cmd i5 holf a subspace DE W321 3? 1m Fortmt SebSPaccs = 3 L} Sobslaace$ CF IE I) Rm, flan? H'WbUfjj" ‘Hfie Origm m “23 is CA Subspace 0? IR? 2.) Hm]- ime eroufl'h 'H'xé’ origin in {P3 is OK SubSpace ‘3) iR' rs ox Subspace 0? Hsehc. 14> {[55]} a seam-e eF n2“ LL) Subpcxces of IR; ,- U Any line Jrhrougk er origin is a ngbsPace of- [R2, . 2 . F 2 2) R is a SDESPQLC‘ 0 IR. ’5) 15 [3] } is a sobspcxce 0F Ra . 1)an 2— : Li“ R be om mxn mC/Cfrix . DcFme U The rm" space OF H is ‘qu set 0F» all Hwe SO‘vChOfl?) f0 fi‘e honganeeous sxfsfem F\>< ‘-'= O i anA i5 Aeno’feé 593 mm“ (A) , I e e . mum) = { 2(6an Such 'Hmct axzo}. '3 2) Image S’m‘ACC OF A (S “we gef 0F (1i) +56 . _ "I _ ' veq'org T in iR suck that Y : RX ‘Ear Sonfiq . _ n - 1 wafer X In R ) cmA I5 Amara! bd pm A) . . 77 , WIN A) r: i H X 3 37M” ck“ X m “2 } DC’F/n 3 [er H be an nxn mcd’flx . Lef X be (Tm 6‘38“ VCA‘Me OF R - Re CCL“ ‘Hfi (kt Lug dQFH“ 81‘! EA to be ’H" ‘9 83+ 0" CL“ ’H1L’ SO‘LQWOYWS "to {kg L100” DgeflEOCQS SYS‘TM ()‘l'—H))§‘=O , 1-6. V n E) z 1’ XER Sada {hut Rx: ”AX/1 .. E) 35 CA“ EA “HG Q gl‘flenspace 0F fl Corresjflohd ‘ (19 t6 9\ A "V7, No’fe; a) 0652 2) Eve” nonzero ve’dbr' m EA is an €s'gt’hv9ctim OF‘ R (or/“esporflmg to) . Theoreml: Let R be cm mxn matrix. Then n i) 'rmIHA) is a. Subspace 01C 3R. 2) imiiq) if; C\ 51156an 0F Rm; 3' Eff} 0} u . Ho goa HUM" D \aeiongs +0 mum/x), HuereFore (Si) IS Stfitgfiegl 0 Le? X‘ and X2. be e‘emen‘t’s 0F rw‘HA) . 'Thean/ fix. :0 and fix). :0 3 Ed deFInd-foq 0F nolHA) ., Now, RUM—x2) = Pam/1X1 :0“? ”:0 ‘ “em ’9er; Eda-03s er‘ now/M , am} (31) is scm'sfr'ed- L9 L2? X. be in mum) , and a be a [ch nufl‘bCr- Then PIX. 3 0, bd AQFimAhon 0F nulHA). But flux/(i) .== UHH X” 7; (xx 0 :: O - Harm (1X, bejamgg t0 nu“ (A), amt (53) is Satisfied . Sm (Si), ($2) and (53) me Sasha? mu (A) :6 a su‘bemce 6»? 1?”, Proof 0 F (2) i5 53‘ m“ 36W (‘EXe (use! ) [”3 'f/heorem 2 Lei” F1 be cm 'flxn (“Ck—tn" / WA A be cm egaenvalme for H "Then EA is OK 5Ub$§pate 0F 1P”, Proof; 0 Ho: 9‘0 :2 O ,. Thei’éfOfe o beiongs +0 EA /ami (Si) is Sahsfieé. /3 0 Le} xI mad X; be demerits 0F E‘,\ p +haf is, HX|=/)\X' CLnA flX2=//\X2_. ThL’fl, flUfifl—Az) .: RX: '0‘ AXL ~= AX..+/\X;= A(K\+Xaj, whmk imphes fivod' X‘\+)(L bebflag 'TC‘ '2) , anJ ($2) is Sahsflec‘ . @ L2? X. be m EA and a be (1 real numbarr Then H (6%; z a (AM : oi ('Ax.) = Max.) , henCc 0X. belcngg ‘tO E3 , omd ($3) is Scrhgfi‘ec‘l- Themem 3 LA” V («1nd W be subfipaéés Cf Rn. 777W) Vfl W is aiso a Soipf'ppche 0? Rn, when» VT) W (.5 H1? mfasech'on of V and W‘ Eff ~ 0 V :5 a subspace of if?” a-—-7 0 belongs to V W )5 a (subs/Dace 0'; it?n s.==> <9 Ema/">95 H W- Hmcg 0 belongs +5 vnw, cma (Si) is :sailpsfi‘ea :- a 1.6+ X. and X; berm}? h: V n VVII Then, Xi bdmgs '“i’o W FIND 21 V: Subspace“; ‘ W : Suhstin, X‘ .4. X2 beionfls 117 V Xy-i— XL b69093 1’0 W '1 henm VflW SQhS’hES ( 52) a Lgf Xi be an element OF VflW J and a L)? (L real nunfiber Then > X 6: V VzSuBPQCG a . e V‘ )( e VDW x7 & ‘ “ > K, “.2? X, 6 W “X. 6: W 0X, 6‘ VDW ‘ Henu (53') )5 Scufisfied - E72] ”I 3 DCFnL; 191k" XU,V..,XK be vecl'ors m R. H vedcr t X! ‘4- "t2 X2 '+ + 1k )(k 7 tum , tk 6 iR , {S CaKecj cm linear“ Com‘b‘hcdpon of X, , , Xk , 0 Th? 56+ of (1“ “(WOW Cagmb.-na"r10n ’5 OF X‘ ) ,. . - 7XK 8. /K [5 COKHEC! The span of X, , v .. , X: 7 0.333 5‘5 denofii’c‘ b3 fipcmafxn‘ngxk}. Iee. spam {X19 "--) XK} 2‘ i ti)“ 1' “'+ika 3 tUJ'v'} 1-K C— IR}. Theorem 4 .: Let X',),..) Xk 56 veck—crs m R”, x ‘15? cm U 5; Spanfxa2~u KR} ’5 a Subspace 0r” WU. Pmof; Bf dQF:”\ 0‘: Span, we k’now‘ U = i “tIX:+~--+tKXK : tl)--~)fK :1ka feed}' a oX.+~—-+ OXK :0 belon‘jS “(:0 U- HenCe U StthFié‘S (sn- a M T} m; E bemg £0 u) ie, T,‘ = "ttXV‘f “"“' {KKK 6‘03 Ya: 55Xy+~~r SKXK t1)“, “W .2 lea] numbers $5,“, 5K real number; Then ‘f‘ _+ 7: n_._ (is-r31)“ + .A .. .f (“tK-tSK) )(‘K beiongs to U , hence u :scdv‘sfied (‘ 82) a 191’ Y, is? in U (hm! 0k 63 (I? . Th en \r, ,: h XH- 1- ELK XK for Some real numbé’ls the“, {K . Heine 0‘11: (012:)X.+ + (QT/K)><K beiongs “H‘- U / and ($3) :5 Sa’flsfi‘eg. €7.91 ...
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