Final Exam - Solutions

Final Exam - Solutions - 1(10 pts Prove that the set{3X2 X...

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Unformatted text preview: 1. (10 pts) Prove that the set {3X2 + X + 1,X2 —— 2X — 2, 2X2 -— 3} is a basis of P2(IR{). 4%th {91(3X2+)(% 7k j)(’)(1w2)(w2)+ c {2% <3) W73 (L 0, (Ba W #626) X (a :2 A )X +~ (542%; ) ﬂ /zxw2é Z 51"29W34 {a {Mr/"ludﬁ "#21; (5M ’5 W6” wag jg; O/ :2!) 4" ’ [VIN/1,! L (may? at W :2 \$951? , Matti/{xi W {7(57‘ a7? JV 2. my ‘ 65mg Wm; Ezﬂgﬂ / Emé’ ﬂ"? if ,3sz TmM‘fm “W t M 4M“ v "/41? 1/372) j’nmg g) fly it {fmz P :15?!“ I! r N f f “I .t > New jijﬂijé} 63W! S33) f iii/5;» ﬂaw/4’] \ ' ( f , INQEWHMW WW5 W {ﬁx/«[7 :3 1/64; “It/iii?) {7" r5 4;: Wm” 2. (10 pts) Let T : V —+ W and S : W —+ Z be linear transformations. Prove that ST is one—tO—one if and only if T is one—tO—one and» R(T) ﬂ N(S) = {0}. ﬂ Afﬁlﬂ/Wé/ I}? ﬁMir-‘i‘wmgq I? w mg) for (W); W, § 77%) 5932; 04:? :52 ffhéé 132’ my; M5 7 I?" M?— * MW (5?; ‘76 ) M (g): J/Iﬂx/lz/{d~ ’7 7"), / ﬁzz/2,66 “1/. Barf W, ﬂ/a’ jzﬁgiz é/W’W) ﬂ/ 57 gaéﬂ/[ﬂ/J EM W677: {/0}? é/kgg/ 6 [5 ﬂ/M/«n/amg, 5; mp “510%? “Xﬁj Tm 0d ‘ 71mm /) MKS?) va/mwé Rﬁ'm 5m 635% M) W) jag/W wﬁmwmww »_ A H; :35 g4 % N f U. TM) 5 W of, (77%)) r: a, Lé’f” ﬁ' TZ’WC): 5f? 51/) ., 5% WM] 7% .A. A Z WM? 7: 0;? {g 2/ ’7 R m NW VWTW (’2 New wé/ MW / {7% 2M“ J’f/lcex I35 _ OWTW/KI WT)::Z€§/ :Ja y‘iaOl ‘ m Z1N6“) M57”) fa} W77 ,3 A M ﬁ/ﬁ‘m’ 53171! M 50 mgr): [7/69 / aw!) / x , ., 3 T )5 W» ﬂ «WAMwWW B 3. (10 pts) Let T : V ——> W be a linear transformation. 'Prove that T is one—to—one if and only if, for any linearly independent subset {211, . . . mm} of V, the set {T(o1), . . . ,T(vn)} is linearly independent in W . (Hint: For the ( <2 ) direction, try proof by contmpostttve/contmdictton.) ck, \ AﬁjM/ﬂo l i)“ ﬁflét‘l’irWM/y Lat [Va 1/ 7 VA to at tiller/‘9 Sﬂ’i’iiii“ {/5 5%00m in. 506%”? fly “7143' If! L n (L y ,r) ﬁll/W; T I3“ {Ill/4i!) (Xi/JV; /l :14 / M 4; 1/; 6 BM who? f is’ moire/267 [W :6}? 4% I/v :4 a / 3 i \ But now 51/1/45; {AV film} If Meat-y gt “We \ I y 7 t ' [W5 T (“L M 7 Tan ) /’/l i S f, n W la Vt Nth/{3%} limit V7 n r/ l; M 7"" WI! 3/ 1’ J m 7»: v a tilt/l (It); [(14) 9:3” \$1 flit: t/ l 55mm T [5 Qﬂ; mgr/‘twm, _ l3 1m N (77:21: 5} 6W“, iii/W :30 «6 ﬂ/(W w/ﬁi \$52 ‘ / 2 -- r. ' ﬂ Won ﬁg gm" adj lg: law/5 m. m a goat 77% /5 Jim“ Z/ My at :5: / Malo {in “6147", An}; mam 5f) tth a? x V .v a l H / ’ Manila/r: I‘l’ ’55 ” ﬁr My /f/t Maﬁa/W; \ [kt/“71A, at l/; [Md/ﬂy 70,, ,3 /,:4, M it W, l e r 5/ NM wiltmfmﬁr/ t‘tzg/ arr/tar W} 2"??? #5" TM ‘ mat to Mr twang/,1, B 4. (10 pts) Let T : R3 —> P3(R) be a linear transformation. Let 5 = {(17070)7(171,0)7(1,1,1)} and v: {2X3+X~1,X3 ——2X2,3X2+5,X2 —X+3}. Suppose that 2 1 0 —2 0 3 7.. [1%— 1 —1—2 3 0 ——4 > a‘hg (’3 57/ (g; ex 512 :2 5. (10 pts) Deﬁne a linear operator on PgﬂR) by T0") = f(—2)X3 + f- Compute det(T). La? ﬁxiﬂ/ X} WW man/Mfr; [7%, «a a 3 w” :3 m x {m w ((5) 50 g 11% :3; " WWI/Wei? { 2 { [M6 é/X/Mﬂ/ [147/7 724/4? . 17571 09 AAA/2,4 / 6. (10 pts) Let V be a ﬁnite—dimensional vector space, and let T be an invertible linear operator on V. (a) (5 pts) If :5 is a nonzero vector in V and A a nonzero scalar, prove that A is an eigenvalue of T corresponding to the cigenvcctor a: if and only if A4 is an eigenvalue of T ’1 corresponding to the eigenvector 3:. /l [9’ cm talus it it; fig 647641 WW '3‘: r r l l/gr: : /l a: <2? T“'("r(%)):: 7‘"”{2l»:x) @ [lac ‘3: ZE/éé) g? /l {f} {M é/jgi/i (9; Z (in/Yi‘égiZtWiat/fﬁ,“ “if; r r / n W (gage/1 1/5/0ng xi,” [ill (b) (5 pts) Use part (a) to prove that if T is diagonalizable, so is T‘l. m H 2 l I/“l MAE; l 5:??? ﬁlial? W?" “all ﬂirty/iii??th £15146va (Tigris/s 5, l) k ﬁlm; (ix/W373 (it If? f {W #7 “if i (ll: l/ mm? m1? 0?” afar}? realm} if???) My é/fjéﬂ realm“ V # 7M6 is; 4 l/ Email/«2:75asz I! it); emit/army TR] 65¢ “the MW; ﬁrm/w WE»? :r/wml ah? M67 J" E . a P ~" 4 ’9‘ M i 14 a f i, i, f /{):;§ 1 f; v w“ I l V i 7 § 6i [7 f, lie; f N, ( I / “4 v l“ a ll 7 15 ‘7 Ml? [7/l«ri,lfrf?;\$‘i l [J 7. (20 pts) Let <-, be the standard inner product (the dot product) on R3. Let y z (2, 0, 1). Deﬁne a linear operator T on R3 by T(:1:) : (m, y)y + 258 for any :6 E R3. (a) (5 pts) Let 'y be the standard ordered basis for R3. Compute [TH {/l/ M! W); will WM) «Wk (2m, 0) (2,4 /) 2am (WW (:W) (Mia) ‘ r [ml/pl]; ﬁll/W) <(ﬁrlx‘lylx (2/3; [QM/U + (ﬁx/M): (/4 2/ W [TM/CW]? fMW) ; <(Mi/l/ (/ng£,f)>/2/p/ ,4)?” 2/45; (2/6; [NM/i}; (b) (6 pts) Find the eigenvalues of T. State the algebraic multiplicity of each eigenvalue. ‘ altar ( (/2 } e (Mt/«MM a, v a (2va ( ti + w ) (Qt-“AKA“QM/l6?) v:anfaww F x x i /’ I j i“ f (LU If}, if] iii/‘9’“ “if ifﬁyfg m it i if a ‘ f of; v 1 K , . ‘ iv Ol/M’l (WM/l d l l (c) (6 pts) For each eigenvalue A of T, ﬁnd a basis of the eigenspace EA of T, and state the geometric multiplicity of A. l (:22 W ml ‘/ «rm. I‘Xv’ 1375/} wigs/ﬁg; { 6% What/é. ﬁm/J/ ’1 C “iffglfffﬁ /Vjt§j/";'/J/fc:iiy r 0 e5, éﬁ N viz 0 Jr/ w “t 0 {jg/7 1/54be 545:; N M {fiﬂéjéiﬂ’lﬂ/él / 91K? ‘3' I l i" zf f C) can/icing (d) (3 pts) Is T diagonalizable? If so, write down a basis 5 of R3 such that [TM is diagonal, and write down [T],3. If T is not diagonalizable7 explain Why not. i p: (00;. 2/ (it ( ,5 it %‘ I/Ia‘ 8. (10 pts) Let V be an inner product space. Let S be a subset of V, and let W={UEV| (v,\$)=0 VmES}. (In other words, W is the set of all vectors ’0 E V that are perpendicular to every vector in S (a) (6 pts) Prove that W is a subspace of V. L” W V! g W2 <02, “(1/3; 2:32 W265 WA <v, m, > {a > +— (is, :;> g) 47 5? (a ég ' ‘5” VI 7' U1 W TM? W Mag?“ fit/2%. / 3, (4%“ V6 W 6M (fit a £62; 4 far/W: 5,7“; (MM @4747 55’ <‘“// X“) «:2 «(m/>Wﬂgg) W66, mm 41/6 W, (was; (/17 {53’ 5556? m (A Lap/mam, 5/7105: W WWW) W 5M 1i?” dab/“m 6W” 1 50/44? ML! 7‘ art/wt I? r ﬂjﬁ (b) (4 pts) Prove that if S is a subspace of V, then S n W = gift/V167 5 ’3; (A M Mm eta/m 5? a? w w (327 )i 59 0g; 55/) w, W; m w 2’ MW (“1’ V6 WI 5;sz Vém 8W7” 5/7: as I 1/6; 4; W&(// (M; Mgr/l5 (3'7 fir/“ﬁg 7* <10 1/ 2 50‘ (/5/ [9i L///: 0/ / r‘ / m {f i I, g 1/ “‘79 Wm 20 M/ g at}, {a /> W’ e :3; as 1/3" F ...
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Final Exam - Solutions - 1(10 pts Prove that the set{3X2 X...

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