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Spring2006Midterm

# Spring2006Midterm - MATH 136 MIDTERM EXAM JUNE 5 2006 TIME...

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Unformatted text preview: MATH 136 MIDTERM EXAM JUNE 5, 2006. TIME: 90 minutes NAME : SIGNATURE : I.D. : INSTRUCTOR (circle one): Khandekar Stebila There are 7 pages including this one. Attempt all questions. ANSWERS should be put IN the artistically designed BOXES when provided, but FULL MARKS ONLY IF PROPER EXPLANATION OR ALL DETAILS OF CALCULATION GIVEN AS WELL. NO AIDS OTHER THAN ‘PINK—TIE’ CAL- CULATOR. 9‘ (a) [5 marks] ﬁne what it means to say that a linear! system of equatigns K _ v11: i9 6 R /? Jug hf/7/fn/ __A - 3 4t Aﬁ ~52 (M o-iéll’ wily/5J4 /t‘hllf'§9(f}'e‘" 9,7,“ I, 71/2? Mam'x 7;, J ;4 I? nuanﬂﬁenf 170/ #m 7mm; éeﬁ” J'th 3 -— I” 6 A76 (b)[15 marks] Find all numbers c1, 02, c3 so that 5% 4 ,' l/ ' \ 5 I133 - 21134 Cl " 7 ‘ 3:131 + 12122 + 1123 + 71114 C2 (171 + 41:2 + 21133 — 1114 H II II is a consistent system. is /’ / ‘/ Z \5 3 \Z \ 0 O i I l/ L A C: / 4 2 «/ C: N a o if {0 —3(,+c ,v o o o 0 \$‘C. - 363 + (Z 0 o ( ’2 c, 0 o ( «7. c! ( G M; ij2[( {5! '3C; + (1 Da 2"; 71‘7 (c)[10 marks] Express the general solution of (133 — 2554 = 0 31171 + 12232 + \$3 + 71L; = 0 £171 + 41112 + 2533 '— \$4 = 0 in vector parametric form. / ’7’ 1 3 /2 l O A ( 7' ﬂ 0 0 'F W ._, 0 O 0 0 Z 0 o l "’2 O 30 \ /—L H 7 / V 0 3 W N 0 0 (9 o 6 ) C) a ( ——1 7K, Hal-f 0‘71; +37CV :6 76,: J/lz «3251/ ac? 4—1217 so 763: wa,’ “ XL: 76"- 76“ 2‘1: 26‘! (d) [5 marks] Describe the set of all solutions to the system in part % geometrically (and brieﬂy). \* 2.(a)[10 marks] Is the set ’ 1 1 3 % {(1H2H4ll 1 3 5 linearly independent or linearly dependent? Explain. H3 1/3 /z‘/~otx I3I O’L'L \OEQ c we 7’0 7% J 7‘» 74M / Wgh/‘ﬂ/f: r a W C’V ‘Jﬂ (SleélarlﬁShojt/lfat I? 4‘! #ll Vﬂﬁ’5lef é/(ﬂ'fe ‘ 17 5065/1? Aave d/vz‘vf 17» €414 1 1 3 ll/MMA zlaqtge I? (Zr/u?“ - SPAN{(1),(2 ,‘4 meta/2474.917 X i 1 3 5 all oflR3. 7? "‘ is not - W {7 5 MN} ﬂI/f/g/r 7% /z»z l/M/g » // Z é k“) a; ‘T/ ﬂh(ﬁr (07145)“)4 233); M W ‘ 0 77/ wt de/hm/ ,4 Mﬂ [a] ’ 7.349“ ‘ﬂu iquZﬂ/é/ 27 I‘Lﬂﬂﬂf/mlq 7L (4 W! ,? [Lil 4, [ll/I 7L /21 (k “Win/PI! Lﬁ/qm h t (4* ,‘7 [kﬁr 74A,”; Mgr-W71.) 7’46 row WMWS 19w} 4v M 7m WMMM, W I? (\x V) WW“ W W M/ 64406 Me Mama/M éa/MW' . .4, F -, all 7/ 1... mn‘ztm/Z, 3.(a)[10 marks] Prove the following identity in R3 : C(v + w) cv+cw. That is, using any facts about numbers, but not using any theorems from the course, show that the equality holds for all real numbers 0 and all vectors V and w in (113.. [This holds more‘generally in R”, but restricting to [R3 will simplify the notation you must introduce for vectors] Give a short justiﬁcation for each “=” sign in your proof. 9 2 c (v + w) — W = c( (WW, , Vzwtvvgwg) /WW4;6 75W : ( C(Vl+wt) 7 “With/2)., Cit/MW» psi/f/[éf’ftgf ( 6C VPk CW3.) C .éa C W5) ﬂﬂjfgjﬁfééz¢ry7 ﬂed! :; (cvmvcvucvﬂ.tﬂm/HWZ’CNZ] // M7. \l urge-7s”; O : diam]? + ((wawws) igjit/x T: C V + CW // ﬁgfdf‘ Viz/75r- £7 Vwﬁp (b)[5 marks] Assuming associativity of vector addition (so that we can write the two sums below without extra brackets), use (a) above to deduce that c(u+v+w) = cu+cv+cw is also an identity in 1R3. £177" 77:7«Lﬁ ,ﬂen we Azw /c(;+tj+5;): 5/5229 ‘ “A754 Axg ééwv Zflvﬂ/f 5527» C5; j #1, 70:) mm: {Am-M am a r out“; “Wm/*5?) > - C" I7 '> /% 35%) Mng- ~ (4+C—FCW ' mm {\W-’ ‘ vH/Lé /Mh//é 54%. W71 WKy/ﬂ/ﬂqu 71, . “W 9 ‘4 14”“th 3c f w . géﬂé L‘ > 6 ~12. «9 7 I? “g g {? r3 3’ 4 {Si 2 ,z / 0 l 9/? '17 y «I / Io «1 if 7 D LI 0 -2 —1 0 i 3 IO 0 ’1 ‘0 O O P { l7 .2]! OJAZ» ‘ 'Z. l0 227 ,0 X 5. (a) [8 marks] Explain why the SYStem ( ('J l) ﬂog/“7' o 0 x n matrix with k < n . must have non—trivial solutions when A is fill/’7 {Mfr/'61 74” 04475;... “Hue “iv‘iwad {AWAXDON {rag W757, maﬂfx ,. )6 MW IMF]: (7 Van/1 4.4/8 (4 WW7)". 14 {T AX”? 4'7/ £<hl 7%”, f]- (ﬂr’l/ ﬁat/Z M [/W/f é!(4/ﬂ7€ f)! an? Act g 2 WI f2 WUCA WW I‘"’%b“‘r M43“ z" ﬁhﬁ/d ’45; .55 A“ 741% MW/Cf, m1 fa 41f {‘nﬁm‘ﬂﬂf Ami/1y mﬂvw/‘ﬂu/ ﬁ/‘yWI- I (b)[7 marks] Using part (a) if you wish, deduce that if n > k then the indexed set of vectors {v17v21' ' ' )Vn)} in EU“ is linearly dependent. /< 4"; mile/a! f&/ x/ 1435791”; iv, ,IQ, ,Vk} m K 7‘; eymvﬂ/éxyy 14v [[6, am 42 ask/«wed an...) a W mofr/‘k ‘ V U” 'k \I‘“ ng )UI ‘. 4 . \VKE 1? Va“ ~ “ z 7. ‘ .H U‘“ v ’l (l 3, i- ‘. 1” yin V “cm \lgu. ' “4* W ILA/019M ﬂg w7hiwz‘r/ Mfg/(74f 7%»! W 74/032 A an (a) MM MM” £41m”? 7‘p #74 Jr Mfﬁﬂﬁ/J AZW‘ 7» m w? W WM/ 7%»% 214.? M4“th d7: ﬂaw Mia/wag /l( 2%; #6 WM 5.1quij W/~x 4/», )5’1‘2. [aVI’zZ/y‘va/zﬁ 74p ﬁe i/LN‘ Hsz W [11’ w M» Ankh/£19 446‘”. ...
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Spring2006Midterm - MATH 136 MIDTERM EXAM JUNE 5 2006 TIME...

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