Midterm 2 16-17 Fall.pdf

# Midterm 2 16-17 Fall.pdf - 4(10 20230pts Let T2114 F R3...

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Unformatted text preview: 4. (10+20230pts) Let T2114 —F R3, T(:t,y,z,w) = (m-—3y+z, —x+2y+3w,2x+y—2z). (a) Find the matrix M F (T) representing T Where E and F are the standard bases of R4 and R3, respectively. T110001 (1 _1 2) J)! T10,1,,=(01>} 11,11 1) 110,0,1,111:(1, 0:21 T (0,0,o,1-)=(0, 3,0 3 .5 1-310 MaCT):“12-03 ‘1 9.1-2.0 (b) Let E’ = {11.1 =1(1,1,1,0),u2 = (0,1,0,—1),u3 = (1,0,0,1),u4 = (0,1,1,0)} and F’ =7 {'01 = (1,0, 1),v2 = (1,1,1),113 = (2,2,1)}. Write the matrix M F (T) representing T relative to (E’, F’ ) USING BASE CHANGE METHOD. Hint: ﬁnd the base chaane matrices P, QE so that M F" (T): P- M F (T)- Q. MUTLM “(”111 111::(1) M :(IJR) 1117317139, Chat/Lead £11m F to F .700,— 1L3 (1 001: 1 (10,11-2.11,1,11.+ 1 .12,1,1)11.V,_2.v,+1.v, (DILIO\=_1'(1,0141+1- (1,4,4):—1.V1+’1-VZ+‘U.V3 10, 0,11: 2.11,1,11+1—1),,~2,o_,11=av, +2.v,+1~11.v, 71-3 0 “Mala METU - NCC LINEAR ALGEBRA MID’I‘ERM 2 Code 1 MAT 260 Last Name: AcadYearl 2016-201 7 Name 1 Semester 1 Fall Student No.1 Date 2 13.12.2016 Signature : Time -1 710 5 QUESTIONS ON 5 PAGES Duration 110 mm TOTAL 105 POINTS 1 (15) )2 (25) )3 (15) )4 (3D) )5 (20) —ll-1 Show your work and explain all your answers! 1.(15pts) Let S = (a, b, c}. Show that the linear transformation T : R3 —> Fun(S) deﬁned by T(13 0': 0) = 2X0. _ Xc T(09110) = X11. + Xb + Xc T(0,0, 1) = 2X1: is an isomorphism. If 5: {(1, Ba} J1m(l:un13\):3:cllm(lpj) \$1? IMLT\: Funl 5) “Ll/um cltmafchTl): 3 3: 0 #14011“) ..{(0,001} ([19113ng 11- IS enoualm +0 \$1191» +hﬁ:Wg 11,111.10 indepéﬂctml ESE/l”- i1. (0;): J1 \ 1:1,,(71, +Xt+7Cc)+/[ ' \$1 1.5 OWL lSOMOr—plalbm, I 1 1 2. (8+10+7= 2519163) LetT P2(]R) —>P2(]R ), T(p(9:))= 2p—( 33). Let E: {101: 1 p2: x p3: 11:2} be the standard basis for P2 (.R) (a) Show that T 15 a linear transformation. VF LXL CID-3 e P ( “2“) 1V 1‘ E [R3 (43"T[(F+cpm\ =2<F111 (#1 =-_ 2119c}+21[#x\=:mm\+T(clm\ / Li‘Tﬁrﬂmd agotrﬂév >61: r. QPC—x] =11". TQM) / 5’?)T 15 0’ LFILW ‘t’Y‘aASfbrmadFom (b) Write down the matrix M E (T ) representing T relative to the standard basis. ”Fa/1);.21 :2 E T(x\:Q.(-><\ =r—2x =£>M (T): 0.,2 O TEXZLZI—ﬂ: 2 X1 E - O O 21; 17111 1 Tat) TM) (c)T is in fact an isomorphism. Deﬁne T 1 :P2(]R) —-> BUR) and ﬁnd the matrix V _ M E (T 1) representing T_1 relative to standard basis. _ _ Name: I ID: 3. (15pts} Find the base change matrix M 5 (Id) of P3 (R) from E to F if E= {61 = 1,62 = 32,63 = 562,64 =m3} and F: {f1 =\$3—1af2=332+33,f3 =1—5621f4=55}- Name: I ID: 5. (20pts) Determine whether or not the matrices A and B below represent the same linear transformation relative to different pairs of bases. Explain your answer. Al H l R?“ maJn—Cx A 3 'Hm Columns am «erLcc’lj M (1,1,0) =(2,L,IO)_(1,3, .1) -—1 0 1 Hana—- Huh-NJ Ql—‘H HOG: owl—I :p/A 'lj_ran5f0fMal-lon/, T Pepmswiul 53 A 0“”le be” an [\SOMOTFl’IlSWJ 5M5“; lm(T)=Span{(‘l,3j4)/(2,9,011111430]§ anal Hwﬂforc ArmlrmCTllzg, :13; M‘Q’ A Far mot-Hm B 3-ch (LO/arms are Ule high/gal— £9} aim/ll + t.L-4,o,«4)+c.(4 o) lamb/o) @gano—l-Czo Czo C=O 2c :0 30"?” 4a Ma’EL a+ lo : O 01+sz T? T :5 a, I‘m. Jrrangf represeni-ecl 10?? E we have AmeIMTﬂ Cil-‘m(5Panl(golll(’1,DJ/ll/M;7—;Ol?f)= 3 4:3» ImCTl :ﬂL4=s T is" an isomoqotsm. ...
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