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351Sp11Sec3mt2solns[1]

351Sp11Sec3mt2solns[1] - Math 351-03 Midterm 2 Name...

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Unformatted text preview: Math 351-03 Midterm 2 Name: Instructions: 0 TO RECEIVE CREDIT YOU MUST SHOW ALL YOUR WORK and WRITE NEATLY. Be sure to explain your answers. 0 The format will be closed book, closed notes and no cell phones, blackberries, calculators or other such devices are allowed while taking this test. 0 There are problems on pages. Solve Problems 1 through 5. Choose 1 problem out of Problems 6-8 and cross out the ones that you didn’t select below. Time Allowed: 50 minutes “- m- —- *- g S S S i S g S S S S S E S smusmmwmw rummwmmi. vw’nmmx’kumAru g g, S g S S; 5‘ S S BONUS(+5 pts) Total(45pts) 2 Math 351-03 Midterm 2, March 9, 2011 Problem 1. (6 pts) SHORT ANSWER QUESTIONS: (a) For which values of the constant a is the matrix [1 Z a 3 E a] noninvertible? A.a=—5anda=1 :A// B. (1 7E —5 and a 724 1 A "\g‘ nm§N exwfihfgm hf: dfif‘t‘fiigi 4‘0 @ a = 5 and a = ——1 ”Oi \ 1 1: C) D.a7é5anda7é——1 Lt 341 E. a = 1 and a = 3 (\vch (":3 "“515 u“ g x: C) (b) Consider a linear transformation T : R7 ——> R4 such that T(x) = Ax. Which one of the followings cannot g 2% § § g E i g i g i 2 g i be the dimension of Ker(T)? $3 (9 2 A is. as my :15 matrix 3:; rantaéfk“; S at B. 3 We \mm w mm C Mantra} + dim ( lmé’fvfl 2.7-} C. 4 _ Am 4- Var-cm :: 1i -- ottm <. $00 m} D. 5 m “If M Vania (/39) E. 6 gm Lg Yankffii} Q: Hi J (34% (¥.€\(u{fi\\) >/ 3 >~:‘> Saw“ <erY§> (Down (be 3, Lb 6; (Os :11 i Problem 2. (6 pts) Let A be an n X m matrix, and B be an m x p matrix. Show that rank(AB) S rank(A). At?) w. am“ map matrix Gm“: 3m (NE) g; 13m an, ? . ~>__ Me “a _ “’ L843 32%: Iméfif‘sfi) 52.7% W, {3 355; TR aajfiimy“?‘?\fl3§ X:{ \V‘F‘CBV £9 Xe szaa :2; m<ae> (5.; Km CA“) :5 am {gmgw} g am (mm?) M743 :5) runaégkfiy‘) g. rmgxfi} 6km; éA‘MC’Lw 5M“) :W‘i ~ X OM~ 3% { mf Ma?) 1 mid AM 4 Math 351~03 Midterm 2, March 9, 2011 Problem 4. (9 pts) (a) Let A be an n X 10 matrix, and B be an p X m matrix such that Ker(A) m {0} and Ker(B) z {0} Find Ker(AB). Please show your work. Let Q e” tam (Afifé) m; $333? 3'}: 3% A (bit a: 8 “ii-“32> £553}: {2“ tier {5%) :3" Q03? in ‘J, ”>2 W B X i: C) «a» SEQ; S “(fix-“53:11,? ramMamam011mm)MMAWyfiflmfieEWW/tmamvmemwfiwmw‘ww ; ad" - 7:3" imflkté‘) x $91}. (b) Let the vectors V1, V2, . . . , vm in R" be the columns of a matrix A and let B be an invertible m x m matrix. If the vectors v1,v2, . . . ,vm are linearly independent, show that columns of the matrix AB are linearly Wmmmwwimmmmmmmwmzmwmnmm independent. i E ’* “2 ”:5 "”5 \ ":5 "”5 A [ \1\ , -9 "Wm WWW?- Vi , A ~ A‘VM are twang i-rxctwgmwt. i i i I E) Om {marinate \mxm waffiy ~17> KEY" < E?) ”“ (2,; {3% «MmNWq-wm wmmmw P25 Wt (as; / iax“ (new ‘23:)? 3cm, 1‘5 goiumn a; 26’ RE (”we imflflj wdfiefldfltfi Math 351—03 Midterm 27 March 9, 2011 3 m1 3 1 ~3 Problem 3. (9 pts) Let A r: 2 ~6 2 5 21) Find a basis for Im(A). M“ M a :33 Vivi: 7.5 E M” i3 1 “(a 2M :3 w 0 Q Li“ ““\ f) L if} ... «‘73 3 “E; v O O L} -\ 1;) ”3 O Q % w , i w; ”‘5 A 505“» $1M” Km (K3 \iu { “25, , 6;] ”f; i b) What is the dimension of Im(A)? dam (m {a} $1". . . \i c) Find a bams for Ker(A). w S Q __ 3 _ w, E; Xllgxz fizm3¥%mA5—t+am~j§” t 1%, ‘Xa"%"%x1 «Hg; m:§xg 3C) ~“‘~‘~‘> H'XfiMXL+mC> :L; xbm W‘s. W. H” 4+ ylxgrfizamfi yd: @632». :r. :«3 A iodéifil [Agar 3452;: ff) E3 ".5 5" 411+ a m “i” , *1 “ii; i” 3 :3. 0 :3) :;"E” a”: my : é Juéwé 63 > i[5]1 if E :65 j“ 5% O L (JV 0 ' X‘, n in , W. g V??? “‘53: am; :7: 5i *3; a“: “mg *3: \ I: V {mdwmaiéfiit d) What is the dimension of Ker(A)? "’ elm QMW Mi; - 3L , Math 351~03 Midterm 2, March 9, 2011 5 Q Problem 5. (9 pts} Let v1 and V2 be two unit (”Vlfl :2 ”V2“ 2 1) vectors in 1R3 such that V; -v2 =2 0, Define : V3 : V1 >< V2 SO that “V3“ m 1, V3 » V1 z: 0 and V3 - V2 5 0. Note that V1 >< V3 3 ng, and V2 >< V3 2 V1. Let g T : R3 —-> R3 be defined by § T(x) 2 V1 -— 3(V1 x x) w (V3 -x)V3. : Find the 8— matrix B of T with respect to the basis 5’ 2 {V1, V2, V3}. E ”'2 E \ B s W x Q , Q E; =1 ETflWE [ ff"? E“? ’2? n a g Q “T x, a; Q Q x «:2 .Q \ l I g M 4‘- E z x E «~3- we; we ._ e 4*: w» W “(‘3‘ M: / m L E 700‘» 2V} “” :3 EVVK X \IE \3 "‘ <V§ ‘ VI ) V3) WW VIE “Ext? ELT { \ICZEEA‘ Q g wwarw WW 3 {:3 m a g .e Q Q .Q Q N r13 e m ’2 - «w «7 E T {V23 5: \/ “' 3 { WW3 W C” ”3““‘2; E “e reindej’vi “We "”9 in?) 1 Q E w EV??? :1 ,, “:3; a A i” E ":3; NJ) “1,, .. ‘ “4“ ”“2? r ”A , .. ”WV \):\12~3({7 w ‘3 _. 0:? av W :2 WWW: ~VQ % Q‘fmx Q 3‘ « '5 2; e 1 Q \ QQ NJ 3 “WNW % «J {Ea M% W “V2. W \ ,. 1 6 Math 351—03 Midterm 2, March 9, 2011 CHOOSE ONLY ONE OF THE FOLLOWING PROBLEMS AND PROVIDE A SOLUTION TO THAT PROB- LEM. CROSS OUT THE ONES THAT YOU DON’T SELECT. Problem 6. { 6 pts) Consider a nonzero linear transformation T from JR” to IR” and some linearly independent vectors v1,v2, . . .,vm in R". Are the vectors T(v1),T(v2), . . . ,T(vm) necessarily linearly independent? Please explain. -31 K "31 1 Wm me \Nmarij \O&W§>%€im\_‘lfi Megan's. \g was “~35 C\\/r+ ,,_ +£2me “:3. C) :73) (:15; ngjg: ,M TX.” Cw‘mm Problem 7. ( 6 pts) Let u1, u2 and u3 be perpendicular unit vectors. Show that these vectors are necessarily acmwmmw linearly independent. knchm¢gg~jmwibmmwm m5 ”‘3 L‘Z’I: H5 ME «4% g “L:\\::2.;:‘Q CE! U‘x‘M‘E‘Clélg‘ ~¥§«C%)K“R% 2:; {1) , Uvua‘: Q ~55 ““5 a M Ma. “*5 q E A in LA? ° <6“th i Caulwgnfifi 5., U1? .0 3 “EX ”*3; ‘ “‘> can? c113 .2: W“ 5.5 5‘ C1 :1 0 4%Y” EQClin lESl 123%" A .. . “A ~-§ 1 lE a my! 5.5 U1 1 Lil we are \irxrzmij Mcxapgrqmb Problem 8. ( 6 pts) Consider three linearly independent vectors v1, v2 and V3. Are the vectors iv1E, 1741— V2 and v1 + 2V2 + 3V3 linearly independent? Explain your answer. A Ax \/ V A \ ~>1 \ N9. N3; (AW \\I\Qflf'l:j MWQEfiE/Sfit ——‘-> C1V1-K—C2QL afagamré 2“” C1: (‘75: C?) :2, @- Cmsszm “ ‘1 «.1 u; ‘ $131 “3!: an.<..,§?1-\l1) + Ola {31*39‘53? %‘¥’% X =3 O ““é I \. . \ . “3 55“ 1 2:15 ($14: d2~ié§1<71+ (—dz+ld,33v$ Ebola V’ «2 Cb “3% $9" Cl :2 Ari; 9;) :5 5‘), sin L2 2:21 Sign 27:35 am We. lfl<iiiil§>9firfilfifil3 ’ Q l W w u 4 arm- Cl wcl 3' . __ dl,’t1_ct\;5::C) “:5’7 &/3 TLC) 51%) Clint 333:0 5% GM 1 f; ’5 ‘3 '3; m G 2:13 timing arr: l’lMflVlifi Mrirfggnmfifi fig“ % E § § % z i; i \Ms«mmmM.(wwv)flvmm‘wmm¢ivm‘nr‘2?»mm)TWWLWAmwwvymmawfimwawamm .mwmmuammmmmv E i (*7 A mmmmm HAWAYAAAA» Math 351-03 Midterm 2, March 9, 201,1 7 BONUS PROBLEM. (Bonus +5pts) (a) Consider a 3 X 3 matrix A and a vector W E 1R3 such that AW! 3 0 but Agw 75 0. Show that the vectors w Aw, A2W form a basis for R3. ~> g: ”“5 €53“ Mk. E?) E s23:£\ AA 1 w x (1" ~* ‘33.: ‘3»? .3) ~35 E Al A5, m :2 a; wx CZ AW +35 ,QA; W 2, A _ A fir-~21; Njf’vfi: fifoi 3’ max At A < Ci W ”3’? (VA: FY33, it: ‘5ng ‘Q A (Ckwx C1 Rm wffi AA, 33A :53 me 6 AA A a A A) A A . 3 4+ M i A» ’ ’AA‘ A \Mark* 21> (Aim—{kadfica/x w VA; :1?» fl Aw “—3.5 2AA A W”: ‘3 \ffifib Eva‘fiax’g amfig Cg Rm“: :2; ma Ca At) “A -"~> .3 1/" 71%) C3“ A‘s U.)- ~31 b V O ' » "‘5 m {arm A633“ fig) AA Ck magma 92):“ AM ’32 . “$0 (b) Find the matrix B of the transformation T(x) = Ax with respect to the basis B = {w, Aw, AZW}. \ n t t E: [Tmflfifmimfifi Ezéfifi :3 w , ‘5’“ h “A? “g: i TN) > 3R2}: 7% iTéW>j5§~$ ; <3 20$ ijsxA Aims: Ea: 453w A ,3 A A > RAT; (£33 EiRiib a” g ¢4§§fis«Sagas?fiéfivaxgaagéqdéiggfie ii; :52: ...
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