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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 1310
MaCT):“1203
‘1 9.12.0 (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 (10,112.11,1,11.+ 1 .12,1,1)11.V,_2.v,+1.v,
(DILIO\=_1'(1,0141+1 (1,4,4):—1.V1+’1VZ+‘U.V3
10, 0,11: 2.11,1,11+1—1),,~2,o_,11=av, +2.v,+1~11.v, 713 0 “Mala METU  NCC LINEAR ALGEBRA
MID’I‘ERM 2 Code 1 MAT 260 Last Name:
AcadYearl 2016201 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) —ll1 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 CID3 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—
HuhNJ
Ql—‘H
HOG:
owl—I :p/A 'lj_ran5f0fMallon/, 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 motHm B 3ch (LO/arms are Ule high/gal—
£9} aim/ll + t.L4,o,«4)+c.(4 o) lamb/o)
@gano—lCzo Czo C=O 2c :0 30"?” 4a Ma’EL
a+ lo : O 01+sz
T? T :5 a, I‘m. Jrrangf represeniecl 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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 Spring '10
 uguz
 Linear Algebra, Algebra, Vector Space, basis, Linear map, Standard basis

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