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LinAlg Test3 [Mar 08]

# LinAlg Test3 [Mar 08] - THE UNIVERSITY OF WESTERN ONTARIO...

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Unformatted text preview: THE UNIVERSITY OF WESTERN ONTARIO Department of Applied Mathematics London Ontario Applied Mathematics 025b Third Tutorial Test — March 27, 2008 60 min Name: — Section 2 Closed book‘ Only simple calculators are allowed. Print your name above and student number on the page 2. Justify answers by showing sufﬁcient work to get the full marks. Solve each problem in space provided for that speciﬁc problem. Use margins or back of each sheet to do your rough work. Instructor 7 Dr. N. Kiriushcheva 1 —1 2 3 [10] 1. Find the bases for'the fundamental subspaces of A = :12 _13 31 51,) O 0 71 3 7 g, L i (a) rowspace R0w{A} \ 4 ’2. 3' " 4 CL?) d '13 :2 (b) columnspace 001M.) ,2 I ”l l gawk 0 ,\ "g 1 ’R'» O O G (c) null space Null(A) A: 4 *3 ‘5 5 [maRs O ”A r 1 R9: 0 O 0 0—187 ‘ Orla‘igral‘boo ((1,) null space Null(AT) 2 T » - '1 0;) locum boa EMU) : {Ci/4,2,371 to», 3/ )3!” Q \oab‘u {5cm camps I —x ‘ / I ’1 \ +‘r :? 0') 5) A: 15+“ : :5 ‘ 2., _ 33+1x i .5 i’ J‘ 3, Jt O I bum fjwr ”kl/UL ’3“ AT: I _ ”t b I 4qu ‘4‘“) \'1-L*O \—1Lto \ -3 *\ KHrRL o —\ l4 'R1 0 \ —\ l o l -l , 31 *\ S 3 9‘3"“: o —3 3—3 R3[-3 o i "\ t R1423 Q 0 : 'QQ‘Q‘) 2 CL *1 + b *9. WE ‘ ~\ \A l o b O l \I \Z Nun w) AM 025b Test 3 w March 27, 2008 2 Student :bé: FOR INSTRUCTOR’S USE ONLY [10] 2. For the matrix in Problem 1: ( 0 (a) Find the rank and nullity of A (b) Verify the Rank—Nullity Theorem (Dimension Theorem) for A \ \(c) Verify that Null{A)=(R0w(A))J- ' (d) Verify that Null(AT) = (Col(A))L . g3 Raw-J1 u): m (Wkéﬁ =0Um (99W?) 3 1 x/ N ‘ Ls. LOU!“ LNWU')):1 V “”4111 L ) My! 06 WM 9.9-9 5) may wwuw): 2+1c4; kwgmmmw%wk .L I .. ‘9 NMKM : LEW/(M) ‘45 L W W 9W magi; wwmww7yﬁo :7 1150M w (7” u ”—55 -[ 2’3)” Ub‘r/O/l) k‘\/'\/ 1—1339(“/‘3 lit] 3 1/0); L01'1’3/#7)D(Uw1’ O/ 7— (0/(/-'b,‘4‘) ' Q.E.Dx in 0:0 AM 025b Test 3 w March 27, 2008 3 i O [10] 3. Using the Euclidian inner product for R3, determine Whether the set {u, v ,w} of vec— tors _ 2 2 4 g 2 *1 1 _ —_2 L 1 ~(mamam70)a v#(0973».%1752 W_(\/6107 6i 6) is orthonormal orthogonal only, Mg or . “L \$-49 4.7.13 ma {GD 51% j .4 9P :L 2., a '12)»? .L. J. ' V‘w-k0;%14§)ﬂ> (47; i-qf-b’nfa) \ : ’Lk—‘Z :0 C C, \ mm mm wmawﬁ' ‘? 'Z @ m _w-.WCJ‘WWMMM}E mmw W ' Mﬁw H L3 Wmaﬂ 16 , A Ni: ”/41 3:3; 75‘ / .4 , [Luii— %D+3'0*3—0 20 PM!” , 1 l 2 a Liar—‘— +1 7 M4 ..\ / AM 025b Test 3 ~— March 27, 2008 x \\ ’20 [20] 4. Consider R3 with the Euclidian inner product. Use the Gram-Schmidt process to convert the given basis 111 =(27_1:1)’ 112 = (Oagalﬂl): 1~13 2 (172,0) 0 into an orthonormal basis. ©uﬁ (If: 0’, =(a,~»\, r) “1M1:JH+IH GJZ / ' 4,} / (015/4) T'(L1éj‘?:3_,%) ﬂ _ éé : (29— : 3-,? ~L 8N1“: “1+5: t3f~J—2, 3 3 3’ 3 1 0‘ ® \7'.— U? ’ ' d ’ image {4‘3 3* 8 W6[ 3 2. " ‘ .K C: “1,3th— /“ 0,2,0)» {Q'QJ (Qr‘l/W '* (%+qu E k 12> 3 3) _G : K’s? % '1] ’7‘" Ill i1! (3) Noxmﬂliéﬂ if,/ “U 3 425v - :. 9; 1’1. J.- j W.“ 47> 4"; 42) .-\ 1—! 1L. Q1:V,_ 4— E 5.,4I’L : (TZGJ%7N;&) 3 3 « uiu 5‘1 F 1" f O '.._3 pl 3 __ P‘ ;’ 13:”? 1 ”Ehﬁ’h’? (”ft-()4? 1m "V5“ W e? / A '2 o a} 54.'“\’12.31-0 a "3|,vg; 0 ‘20) "€1.ng J— ‘ G . a 3. ’J— ' (' ’TF’H (2,4,1) (9. ,1,—3L) (51/4, 1) (1" ’ﬁ’%‘) [ 3 1 3/ 3) u ‘ \ 3 3 r. ‘51 +1 ”.5: I A ’1 "— ~ “21 -,\_ £3, g3 3’2, 33 2’ 2’ 3 u u n :0 \ w 0 \ H o \ ...
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LinAlg Test3 [Mar 08] - THE UNIVERSITY OF WESTERN ONTARIO...

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