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# examsol - b{lﬂpts Evaluate L112 and Ln[23 in the farm a...

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Unformatted text preview: b. {lﬂpts} Evaluate L112 and Ln [23} in the farm a? + ﬂy. La [11-3 > 2. (25pts) 3. {15 111125) Find all Sﬂluﬂﬂﬂﬁ to 33 = -8i. uti- 3. {2511115} _ a. (lﬂpts) Shaw that u {3, y} = e‘ rainy is a harmonic function L1: = e); sum UH = E.“ ﬁ‘q U1: {3" m1 “#1 = - E‘sﬁ-‘l .—-—\. é @ m - W1 6) :3 UK :5 him-Murc- ay”+:(h\$y}lsananalyﬁcfuncuun u: 31:32 1-111 Va 19H w: M w: by Uni =- 19.1 V? = I” : W: V? =3? 53-"- I91: 1:: 5 {up-H's. =2 MT=—'P‘?’ z) a, 12’; avg! c. [Factra uredit: ﬁpts} Write f {2] as a mutation of z alt-rte. 5(2): agtt’512+£x‘f{ r— 3(1‘ff1)[email protected] 4. (10pm) Show that Ed dyﬁaihhnﬂ ~ m22+singz=1 E. *t‘i {‘1‘ + (3362- : W (1 2. Etﬁhc-I: ‘E‘nl = 1‘- cvs-z = :3 4 1.}. -352 .: '6’. '1‘ 11- e. A {a .41! " {aﬁxﬁ :- LIE ‘- ﬂ ) *4 . all; ‘2. + .51“ “IE _ __________H___H “II- =) m {’r-l' Inn 3; = i' 4:- .r" a. (1Bpts}Fi:1:i a. basis for the WWW of A. 1 I 3 Lark?“ * l 3 [H41]. 1 '| 5 ( 1 ‘1 I {11:3} 0 “‘4 '5 f? 0 4+ -r j _. 3 -1, a «II- *5 a a D 0.. {10pm} Dues the ayatem AX = B, with E = ( many solutions are there? 2 ‘3- I 1 ~23 “3-. '3‘ 0 *‘f' '5- *5 1 I 3 3 '11. -+ Y-l- "+ 3K5 :5 H1 . ﬂ 1 D 4;. Hr *E- {g} “5%.. r315.) :: —-E a o D a 2. 2. 1 ”33pm” ‘9 Z r! E -1 - u l "' 1 #5 r2. ‘1'" 1 1 ‘1 [514;] ‘ D _} a Z '4- Z. '4’. [3|] puts] Let 1 i} 1 1 A: —1 2 , H: i} —2_ . 3 E 3 U a.(12pts) Calculate the following or say “not deﬁned”: A —t— B, A + 35", AB, BT41. /‘ 6 FEW 444 4:14 44244 4mm. b. (Bpts) Suppose A is a 4 1: 1t] matrix with real entries. We}: of the fellow-ﬂing eeuld he the dimensions ef the raw speee R {A}, celumn space C {A}, and nullspsee N [A] {You do not base to juslifya year answer}: 4 ﬁim[U{A}) = 4 dim{U[A}]— _ 4. 41444wa = s dimiﬂlﬂli = If} MIRIAM =3 dimIREHD =4 dim{N{A}]— — '31th [21)] = T ﬁimliN [All = T 4. 4440:: (4}) = 4 4 ﬁim [42(4)]: 4 4441mm): :4 ﬂim[R[Aj] =4 444mmn= 'r dim{N {4)} = u e.{1ﬂpts] Give an example of s set of three diEerenL veetels in R3 whose span is tum- dimensional. You ﬁn game-mamausﬁfy- yew—enamel; _ ...
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