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108APracticeQuizD

# 108APracticeQuizD - Name 5’ V J Mathematics 108A Practice...

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Unformatted text preview: Name: 5’- V J Mathematics 108A: Practice Quiz 13 April 16, 2010 Piofessor J. Douglas Moore Recall the following Lemma which was prover: in the class earlier: Linear Dependence Lemma. Suppose that (w, ,vm) is a linearly depen- dent list of vectors in a vector space V over a ﬁeld F, and that v1 aé {L Then there exists j E {2, . «,m} such that vi 6 Spen{v;,. qty--1) Moreover, Span{v1,m,vj_1.vj+1,. vomeanmwvm)" Assuming this lemma, we would like to prove the MAIN RESULT of Chapter 1 in the text: Replacement Theorem. If V is a vector space over a field F, (ul, . , o , um} is a linearly independent list of elements of V and V is the span ofa list (W1, ‘ , . ,wn), then m S n. Idea of pIoof: One by one replace elements of the spanning list by elements of the linear independent list, renormalizing to the same size by means of the Linear Dependence Lemma 1. Czaer out the proof of step 1: Let Bo w (W1, .. ,wn)n Show that we can replace one of the elements of BB by 131, thereby obtaining a list B; of length n which spans V and contains ul. Sth—a BAG S?Gv\‘\5 V) {:1 6 SFQV“ a .-- Cit/h) QANCL a). W , ' _,_ “’7' tufWHV-th) 1.5 lmeerlj CLEPC’IHdLEVV‘b . u: 57:; O 50 like! Llheox Depuxderce Lev-nww» implies a w? -—'r -’r . '4' E" :77 "a" "'4. w} é w‘ 3.. lw“) Swot. "blew W} a Open. Lemma“ ,wyl) "r my ~9- wr nua- - oral. 8?5‘Y‘lrl’bl}wn‘ swjullevfl 3' th) x 519°” B0 '3 w? -> ——>» w? mp Let B‘ “" Lu|3\"lj) 5antnwi+i) llth 2.. Cally out the proof of step j for 2 5 j S m: Suppose that 835-; has length n, spans V and starts with (111, ,. . ,uj_1)‘ Show that we can replace one of the elements of Bj_1 by 111-, thereby obtaining a list Bj of length n which spans V and contains (111, . . . ,uj). W V»; ~> w, Ban! = kl)" 1‘ " 3 ml”) will! b‘ onflhl} ST- oehE \l. W? at my __,. ___.,, Hay-Uni ULJ 6 \$3?th (Mi: ~3Uj_g3 OHJED, JbJ'ikh‘} oh& 3 lance “ole”; Llhear Deoewoiehce Ll‘ﬁl'nrntx lmFl‘aes 3 "r I \ ' wd{k\ a \ :2 la; b h) guclﬁ. 'DlﬁﬂL’G m? "a r, *4 Wdlk} J S?C"l’\ (u'i; "gut!" h 3 ' Jwi{k._|)) Ovl’LOL "7 A" w ":wodlnl.) "1 Slag“ Elhl : \f. a n.» mill) )\‘I1W6Cll¢‘lll\rfnlll¢4'll§' ‘ m? ~—9’ .47 -——> w"? m—B' Leo 83 = W. y-«Juj , Wham ., « ,Wetwu Annual, «‘ Whom}. After step m, we have added all elements of (11;, . V e ,um) and the process stops The ﬁnal list has n elements and contains (111, n . .,um), so m S n‘ QED 3. Now use the Replacement Theorem to prove the following corollary: Corollary. If V 1'5 3 vector space over a ﬁeld F, and (u1,.h.,um) and (W1,l..,wn) are two bases of V, then m m TL In” my ' ' Lu? au-qU-Ml \WOLLF-LVVMDL-{n't .\ “7 :0; i 'n‘ amok LW; 5 . to 3 WM“) sinwi «I m r y 37 wﬁ m. _> F- I ' _ «a. twa a " 3W“ ) llMJ—mrlj moLLPmAel—e» o (2 M w? H “5' \ SPELch V .2"? V1 ...
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108APracticeQuizD - Name 5’ V J Mathematics 108A Practice...

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