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### 2010m2

Course: MATH 120, Spring 2012
School: Middle East Technical...
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of M ETU Department Mathematics MATH 260 SECOND MIDTERM EXAM Date Time :22.11.2010 Duration : 10A min. :17.10 Name Student ID Section Signature 3 Quetions on 4 Pages 100 Points SHOW YOUR WORK Q.1. (30 pts) a) (5 pts.) Show that the foliowing functions x,I*8, r* sin&quot;r, fro - t, and r*coszs defined on IR are linea.rly dependent. x + ( t tx) -(x a sfr'*) - fi,n bsux) b) (10 V: -; i: j', I ;I...

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of M ETU Department Mathematics MATH 260 SECOND MIDTERM EXAM Date Time :22.11.2010 Duration : 10A min. :17.10 Name Student ID Section Signature 3 Quetions on 4 Pages 100 Points SHOW YOUR WORK Q.1. (30 pts) a) (5 pts.) Show that the foliowing functions x,I*8, r* sin"r, fro - t, and r*coszs defined on IR are linea.rly dependent. x + ( t tx) -(x a sfr'*) - fi,n bsux) b) (10 V: -; i: j', I ;I =a, (g,t.r6*)=11'=o pts.) Compute the dimension of the vector subspace span{( -L,2,9,0), (5,4,9,0), (3, 1,0,0)} of lRa. t;- G,,", ?r""l'-3. y g ffi 4 ; bJ fferLu + 3 oi wu"L' c) (15 pts.) /",*(V) =z oo o) Find a basis for the row space of ,4 and find the dimension of the row space of .4. tooav lo t -,t o lv l6o zo t f'r lszs: t J" [-To 1\- lll;llffiEil\\lH' et2 t,\ {(, ot or) , b , -t t o) ,(ooz sr), (ooo' '\ l3* d,t*q-h9ruh ), - q.2. (35 pts.) a) (SaS-10 pts.) Evaluate the following determinants (no pa.rtial credits will be given for an incorrect numericai answer to the question Q.2.a)): 12042 23456 02021 00030 L2041 r2345 02040 54321 00012 0003 b) (15 pts.) 11 z =ofr;WIi=(q)'li,i I {rl'l ,':l-'l :ep)(r. (t) -r' o) = 2 ( L{ = 4 li i?\ t;?\-'ll; Compute the adjoint (adjugaie) r o: |l-111.1 l2 ? 3 1 r-sLz{l I of 'll; I ,td'() = \3 l= ! I'lr-['l zO[=-J a ,;air'li 3 ?,,=("')'r Iti'ul =-S at2\'/ rqal la'' u''= tr [lll= -s t,f.'li o1z ez,= li, Ll=-7 a,{- l;ll = 7 a'f-l;i\=o Qz3=lill=2 c't-- Uli=t er,= - A,=fii:il ogl+=*E-;]1 g.u^)- ,4= *'a?* ,rrn) --t^l:;'",tlot = zl','ol ,, (A)--ttt -o,\= h3'-i -tl;il= WqA1;ilfr??li;:1i,1'*Elih:';1'13' -ti:!J*??)ff ;Bi'Ai+fl*l;il't?il:r,'{!,1A' d A'=(;) rl'fl:i]= EiA c) (saf-10 pts.) Given that \a1 blb a2b lL+a a-fl a*2 b+f brb =21 s le 2a42b 12 ar3 bx s2 ar 2r2 : 8tZ 7 a z q( x4 brg brs = x'x2- ,t l; lq I 2 9, compute the following determinants: 4l t*81 lr q el=2.3= i''l=zle L l8 4 / l" z dl {i = 6 x lral' e\= 9xt Gl=yu, 4 sl \ la la Q s I Q.3. (35 pts.) a) (fO pts.) Exbend {L t fro o o lo r l,0oo + r2,r Bas,g - r,3} to a basis for the 'b { {,,,',,-r', o lo-r \-/ Jo N\-l \J J i\ * d Lt ? Lru) X vv b) (L0 pts.) Find coordinates (the coordin-ate matrix [u]6') of LL: r - s2 + as with respect to the basis C : {wt,w2,u3]1 of the vector space il/ : span{uhtl!r2trl3}, where il)r : f, * t2, W2 : fr - X2, and Uh : r + X2 + 2r3. t= wrt x3 wr'r : - !*, J w'* * = !(w;w,) .+ =X-X+x 1-t I -, -- /rn(= | .L-Jr-I1 L, c) (5 pts.) Let to e W be such that lr1, defined in v,/ b). Find tbe polynomial u. : Ii ] I I ) where c is the basis for Compute the transition matrix P : PB-c from the basis B for W to the basis C : {tlr, wz,wt} for W defined in b) . * w. ) lo1 ,= it.l/, -4a1' Lrrl ft"t" cz 9*d su&L,ou t o o f*fR= ro 2 y7= w, I I ,-l Lo0 \-/ v W, wz I ( L ]I/ : {r,x',rt} -4 + bJ; F*1"til;il, r[r1r= ?: L0 -o I;rl ltii ri' -t *, + 2(x -v') =3x - v2 = l'w rt zwzl o vr= k+x2) d) (10 pts.) 2X = w, space of ;_ L-t h] I o 00 t/1 ; ;,-v" \rZ ,"?:Jryqiz' 'tli'!i wilil^fl;,i '33i\-'E,w
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Middle East Technical University - MATH - 120
Middle East Technical University - MATH - 120
Middle East Technical University - MATH - 120
Middle East Technical University - MATH - 120
Middle East Technical University - MATH - 120
Middle East Technical University - MATH - 120
Middle East Technical University - MATH - 120
Middle East Technical University - MATH - 120
Middle East Technical University - MATH - 120
Middle East Technical University - MATH - 120
Middle East Technical University - MATH - 120
Middle East Technical University - MATH - 120
Middle East Technical University - MATH - 120
Middle East Technical University - MATH - 120
Middle East Technical University - MATH - 120
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+=0
Purdue - MA - 303
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