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Midterm 1 13-14 Fall.pdf - ‘ A J-L Défit#5955 331116 1...

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Unformatted text preview: ‘ A J ,-L -.; ' Défit. #5955: 331116 1 2‘2": 40 _ fl-‘izé‘afian I mars-Mics; F _ u m. m n "in 3 U « ._ WWI w A i Q? (1% p.) Let V be a vector Space over R (01‘ (C), and let U __<_ V be a subspace with its complement W : V\U r: {v E V : 1} g U} Q V. a) Is it true that W is a. subspace of V as well? Explain your ;mswcr. skim}; Owe uxwe“ CCW‘GQUGEQ {:M O"! ¢ W” JCEMM‘EJ XVV COM “va 1”; CL quflme (3% V b) if U % V, construct a nonzero SubSpacc X S V such that U Q X :2 {0}. __, .x u .0: -.f X:Sa&£%<\/ m» Xflw w Xfluxicjfiw ueYflU 25> wax"! £0? 50% AGKRG‘LQQ; ”ii “A“? C? Jig-En AV: xiii 6 Lima CGWIETEJAILWCW (ET? (3;? p.) Consider the meta“ space C [(1, b] of all continuous real functions over the closed interval [[5,5]. For each n, the set ’Pn of all polynomials on [51,5] is a subspace in C' [a b]. Is true that their union X "2 Union is a subspace in C [ugh]? Explain your anew-zit What about their intersection HR 3911? Mole H04)? Pa, 4" pm gpmximj 1L. DMHWJ g. mimic >< {3on wfiw 1‘» ><+Y Q X 01’ X" ‘5 Q §Ql9$f7cx¢e Q3 (75 p.) Consider the following vectors a1 2 (mlfiyfl), a2 :2 (—4,D,—1), a3 2 [2; W3, —1), a4 2 (2, Mg, 3) from the vector space R3. Find dim (Span {ah a2, 331 ad) 2?- Explain your answer. km (A1 at +12 Elli, +13% :3 "ELM ewe 1W “)1 «(+ng +&)S:O NMQW oinWfiSPaellQi/a an;ay fiej)——3, {ii (iii; 3),) Let 3 : {1,2,3,4,5}, and consider the vector subspace v {f 6 mm : m 2f(5) ,w) = «4) ,2fl2) 3f (3) ,m) = 4f (4)} in Fun (5'). Find a basis B for V1 and dim (V) 2?. Explain What does V geometricaliy mean? Rib 49(5): RTE. 19(11)::322 ifiegpfigm: - 4M ) AM) 1 ##9##) : 4 g 3} gig) __. 3%, {(31): 3&1: TM 1‘, L’ ago 535:: 33.) Lot 2173 (R) be the vector Space of all polynomials of degrees at most 3, and pl (51;) :r: 3 + :1: (9 7 4:3 ,,, 23:2), pg (:5) :: 2 --:— 3:1; — $3 and 333 (at) r: 2 + :2: (m2 vi~ :72: i 1) the elements of 793 (R) "fest whether 9' (3:) : (3.202 — 2):1: 6 Spam {p1 (x) , pg (:3) ,pg (35)} or not? Explain your answer. Rat ElVCM 'A Fax) :L f Mm 4- 613300 01.. “3x Jrsxs: 39+ fixwxfiigifi) :ffialsx‘vfi) +96% X +2xl+x3j ‘3') +&j« JFSWIO 9) +3f~ — 6‘ :4, {:77 3:17 v—W +39:0 5.7:), :7.— ngyflfi+6=53>eg f 5 BULL Here is QM axl’na WWVLIHM '3’), +§4JPK +31§10 Car-05 3:1 ¥§~7(r5) +o’Q-Q'zl :36 Cl: \che Cl“) 6% 3mm»: 5’71")» {ml ...
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