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Unformatted text preview: ‘ A J ,L .; ' Déﬁt. #5955: 331116 1 2‘2": 40
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._ 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? JigEn 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
anewzit 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..
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 Winter '10
 uguz
 Linear Algebra, Vector Space, Complex number, Vector Motors

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