MATH201 Midterm 2 2007

# MATH201 Midterm 2 2007 - AIBI BII BIII BIV Total 10.12.2007...

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Unformatted text preview: AIBI BII BIII BIV Total 10.12.2007 Name, Surname, Studen No: Recitation Section: Math 201 Midterm II (out of 70) Duration: 90 minutes Part A: Delete as appropriate and /or ﬁll in the blanks. Explain your answers. (4 points each, no points if the reasoning is not given). ~ I. There isésliﬂa, 5 x 5—matrix such that rank(A) : nullity(A), since hawk {gitlrrmﬂfiéﬁgk ,7 f: .5" X]? writ 1114) ﬁdl’z’lrﬁfﬁi) fanw'f‘i)‘: 3’15 we f-‘D Sﬁlg’gﬂf a mﬂfaﬂcéen H. The span of {1,12, 1 + 332, 1 — 3:2} has dimensionﬂiﬁu, since 4/ - I 1' -' ’ I .4 4 ,7 >13; _{...._,w 13m km}? {)2 [/neorﬂ)mbwﬂ?9aw (4’ {9.34.5 X y * are. Urﬁﬂr’fgé iii/fail? :m die/rag ﬁrst! III. Let A be a 4 x 6—matrix. The smallest possible value for the nullity of A isnXQN, since VOHJI’I’fN if {iii}: 6 .ﬂﬁf rank/ﬂ} awmﬂmosgﬂ IV. The subset S of P3 given by S = {p 6 P3 : 12(0) 2 0} iS/isﬂfnoﬁ a subspace of P37 since (here show your arguments for S being or being not a subspace) fa ,7: rim! 52%)ch a‘me SmL’ér a ma £935; x2?/ 04%» pavef >0 :5; ,e y r‘ / l S“! J ‘ \ J 3/, «kw it. by; (“f/LAP 2 9x J? {ﬁe/fﬁﬁm/m ’7 ) Part B: I, Let 1/1 = (1,—a,1,1),v2 ( Lori—2, 2,01 1),?)3 m (1,2, 1,a2+2a 3). Determine for which values of a the vectors v1, v2, 1);; are linearly independent and for which values of a they are linearly dependent. 6 points ~[/1] KT] V7 are [I‘mr% A4 -f‘,:\2 : 0 \$3»; a nonl‘er/a»? QDW/ ﬂaw Mid? E darbﬁvgﬂﬂﬂﬂ .2 f ’2: 4 a A; O ‘ Ki 7 ZN I; 4 5;” 5r (—3 1’ < 4 "‘4 4 ,"Zgz M, .4 ~4 4 23 w 0 i. Z 523 a) O 2 £+Z ~=~—-—% ‘ ‘ ﬂ wig: gig/J O , ,2 IQZQAQE 0 2 £1“; £1 t? I « / K0 3“ 532314 R“ ’4 o 5. 5:25:41," % / ; -l r i j/ ﬁ/ 1 o / Z #3. 0 / (91: f c) 0 £22 m—(ﬁzﬂg 3 o (9:14“ 0 0 £244 0 .. M“ < warm 3 (Aw/7 ﬂiwﬁomoo it?” ( [Em £2,010,731 ) 11. Let 1 1 2 1 0 0 0 0 3 —3 A _ 1 1 2 2 ~1 0 0 1 0 0 (a) Find a basis for the r0W3pace of A‘ (ii) F ind a basis for the columnspace of A. (iii) Find a basis for the nullspace of A. 13 points (iv) Find rthA), nullitym), OD ’ (200 *‘a r"_ 5 i 2— %,Q2 30 lg (xii: @0000 g D n J“ 0 Q 0/ gamed/g .\i‘i0.0~/ OOCDQQ 000C?” QOQQQ ’7‘??? , , ,I @wjﬁyzrjrgyowgfoa 5 {(il/IO’QI)/ (0/0] ’O’O)/ “Mp/Lin /) \ ' k r 130/ A“, O "f a] \ . I ’ W1 y;- :3 ﬁWﬁrle WWWW 5 I? ‘ 9 )2, \$27 >< : 0 c) 1, \ g . o i I j 745:“! >493“ i (b) Show that v = (“1, 3,0, —2, ~2) is in the nullspace of A and determine the coordinate vector of 11 relative to the basis of the nullspace of A you found in (3) (iii), 5p0ints ,__r ’3 , A{ 5 )1: 1:7 Cf/z‘olfiﬂg ﬁﬂgéfﬂm :if E): I ll. ,4 if u e ‘ "’ O _ ,ﬁ‘M’Z Mi I) V 0 as? 11/: 3/ 2 {Kg ‘3 Va 0 fflm‘; -:~ I h I 0/ (1:) Explain how you can obtain a basis for the orthogonal complement of the columnspace of A (Do not calculate) What is the dimension of the orthogonal complement of the columnspace ofA (explain!). 5 points x a ‘ <3 126 v 001%: mm 317“) “1'7 74 " m “J “F I m Oriﬁcpgqﬂaf of fowsjﬂm Val/1V: a fill/7:; was; A; 2:7 ramming a £0,920 [29? lg naz’igfwagf/qﬁ Afgaa 4 /” army“; :27? (OM; 'r ﬂag/XE) I347»): " '3 «F ﬂuff/£3 6447)»: 4, W '7” .\ ‘ 5:! ngéf‘EWWﬁfA £110 diWﬁ3/GGI /f III. (a) Consider the subset s = {p 6 P2 225(2) = p<—1>} Of P2. (1) Show that S is a subspace of P2. (ii) Find a basis and the dimension of the subspace S of P2. . a . . , 7 A (I) - 3 ,5 m emka I 06\$: gym, 0 r2. ,) s OH): O 10 Pomts .. <E§6Q£€efunlm SVFPOR 73/? 6‘“ g x) (Pl/ﬁxfﬂﬂjgm é: . ‘s q(z)-=<;{L—J)é=f£’ =7(?¢4)(2—?" PIER???” gm“ :7(77‘*<?)(293/77WV”’] (“WW/'1’) ; PM “9"” 9 6M :7 79+? 6’5“ - Srkogmeg anJm gmlar mwfiipil’coléoh g (‘5‘: f; :7 79/2} 5 VILUEir—S: 1:7 (’kpjfzﬁ iii/912)): M: =3 (“QC/S (Isz/L‘ WNW “‘3 (m awaxﬁgﬁjf 5’ 4:3» \$70er?? ﬁshy; :: aa‘“<>?4-nsza (:57 3041*3mzn O (=7 .02: FQ/f .— ‘L rm ,7 S: {ﬂo'mﬁ’f‘w /"9"01’536’K/2 t. ﬁe?" 04X’V4X My»? ; x90+ 0:4 {'xrxz) \$0,494 6%: ’- ,._.— ‘ J‘s—.— ~ Sec”an Urge/1424M ﬁrm w ‘ S msigm [5’4] xv}; £3.46 (/1) :5 S M :19 \$113?qu (b) Let U and W be two subspaces of a vector space V‘ Show that then also the intersection U 0 W is a subspace of V. 8 points (make sure that your arguments are complete and precise) (E) (“M W '73 WWW 53m“ U and W’ 0% we 311195578263 of 1/439. @453 éaﬂtfenz)? {a We Vector s? 0 E a” [A/ \ Uft l2] {S ﬁaxgaﬁtén QJJM‘G” 3 M {EU/1W and: VEUGM 9! way one? vé'M "‘w (M (Lt/end? V6 W ’77\ Siﬁce ‘4’? 5/” §Vé§pmc>l7 V/ «£5 aﬁgen Qgﬁfﬁxé’ﬂk‘] as) M/VEM M52} (“ﬂ/é (A; uer 0/; b/ Wimﬂa. 330m 0V5” WMMQVZYM) LYN/6 W > My Suffix?» V w [:5 (Mia scalar ngﬁgizalme: ‘ (In) {14 W . i 5:? M51} 03% £16” W~ Since U, lax/Gift:— V’ )- ﬂ \ ‘ 9 (4, ~, , 'nén yabrﬂV/Mpﬂmfml 0&2: kué’Um’Z 1m 6W j“b’5ﬂ‘>”“mf kr/‘lzmﬁ’ (Mu \$7 la: a? w ’3 IV. Let A, B be matrices such that AB is deﬁnedmmé columns of B are linearly independent and suppose that the intersection of the nullspace of A and the column space of B is Show that (AB)3: = 0 has only the tn'vial solution. 7 points (whﬂwﬁe (Ag) {7° 2 ’4 Hair/i " O ‘ 3:) 751)» 5; ﬁuifspaa A.) V I r 7 (ﬂ iris/E/Mé‘gtr 6' ébélmn‘sﬂooa :9? if) it> gay: 0 i ‘ t I (Z are; 3‘ i i. 4 3 ‘ #3?” £9.30 5.ncz#8(;%mﬂgo7pp kul 20‘ 0 / Li'ﬂfyr ...
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MATH201 Midterm 2 2007 - AIBI BII BIII BIV Total 10.12.2007...

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