midterm2_practice

midterm2_practice - Sowwious W 2 NAME SECTION MATH 21-241...

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Unformatted text preview: Sowwious W 2,. NAME: SECTION: MATH 21-241, MATRIX ALGEBRA Fall 2009 SECTION 1 (11:30—12:20) EXAM 2 TOtal: 100 points 0 You must show ALL work for full credit. 0 If you have any questions during the exam, please raise your hand. PROBLEM_ SCORE PROBLEM SCORE #1(20pts) #5(18pts) - #2 “0 9‘3) - 1. (20 pts) a) (3 pts) The system A13 = b has a solution if and only if b is orthogonal to which of the four fundamental spaces? bECU’Q :3 lo oflho%ond l1?- kit—WT) b) (3 pts) Is the projection matrix invertible? Why or why not? :P‘W was mum’s.- W“: P _ 1:99”‘w3e -. ’9“ ~—————-——— . . H: x {Pl P —-. C) (14 pts)'Il‘ue or false, justify your answer if false: i) (2 pts) If the vectors 3:1, ..., 53m span a subspace S, then dim 5' z m. Rim : Hug mid alw have tr la MWWM. ii) (2 pts) The intersection of two subspaces, of a vector space is empty. m: m m Vto'lvr rs m sum luloéfaee. iii) (2 pts) Ian: : Ay, then'x=y. fig = +5110; A so. iv) (2 pts) The row space of A has a basis that can be computed by reducing A to echelon form. TM: _———_.., v) (2 pts) A plane in R3 is a two—dimensional subspace of R3. leis .. Onlxj plams - tlnnmglc HM vh‘cbah our and: new; 1 vi) (2 pts) R2 is a two-dimensional subspace of R3. 75FGHK: {R1 {S W 0L VLd‘orS UAW-h M- Comuvovwmf ho+ "HQ/Ye,- vii) (2 pts) If v; and '02 are independent vectors, then 1;; and '02 Orthogonal. Pam .' Jratua \n h A E) \/ :3" u C‘ I: \/ 2. (10 pts) Find a basis for the subspace of all vectors in R6 with $1+I2=$3+$4Z$5+$5. K1;=—X2_+K3~H(L‘ 3 X6: Xa—qu—Xs'. K:- Liulia. Kat. XL“ X9. X5.) I$(')C1+ 3(3'1' XL“. )(21 Kat 3(th Kg, K7) +Xq—x‘g) : fzc—MA. 0101010) + K3('{lol I 10.0;1) + (“LL 01 0; 4,010 + its—(Ot 0,0 to,“ —1) : -—— ' . .——_‘i_8 liq; OIQP(O)\ (Mott! 010t{-)1.€LOIOJMOH) 1-(010'0‘0?h_l)3. 3. (10 pts) What matrix P projects every point in R3 onto the line of intersection of the planes at + y + t = 0 and a: — t = 0? «fa-2+ 9‘ =6 {rifle} "ELL—1‘4) :3 Q = (’11-'711) m Q‘QT .‘ T 1 ‘ . F-Qra ,Iaa=[1w2u1[fl]=1+qflzé . r 1 4. (10 pts) Find the pieces x? and In ifA : [ 0 0 J and a: = [ 3 J . 0 O 1 -“‘ - c . . c 0 0 in 314‘ am) Anemia) ll 'D-{mmPo H = Uh 0%) (“way—(l) =Ci(M—\)+O 2) qua:cl —)Cl=2—. —Q:I-C‘ :3 Q:C‘ 5. (18 pts) Find dimensions and bases for the four fundamental sub— spaces of ° BCUL) : {U.o.c.ol), (llama), (.b,2~.0fti)3 3 dim can: 3 '9 Pam) =. {Ham-5), (calla), (0.0.0.03 : sum 12mg. 9 MA) = {x [Ax =03 Gl—tllo-i—Sd :0 61+?Jo:o c: u, 2‘0 +Zc~r26\:0' “:3 unfit _O n- Who ‘ w LZ—b' =3 MUD I= {(-Uaibi -10. 0‘) \bglszj' : {M «Lip i1031b€ii33 Ewan-.K —ln-’h~!.c')7> 2 CHM tempt. flan-.0- , 3Q-+z'o+ttd=0 =>cl=.o. : : o‘Ioli‘Oy‘Cf/flflg :7.) BNCAT) = {Loioih‘033 ' 4 dim MW): 5,. I o o o q '0 Q20 ‘ Z 2 0 o 5 3 O 1-) Zanlozo =)\o:.o. o ‘L o o C 7 0 3 2. 0 L1 0 6. (12 pts) What is the matrix aSsociated to difierentiation-on the space of polynomials P2 ? Find the nullspace and the column Space of the matrix and interpret them in terms of polynomials. "T'- 91 “’91 W2. « {111.ng Tm :oen (Ha-x 0%”:— O W B Tm =l-A‘HO-xfl A: 5x0 '24] ‘ .P') T'Ciljrlfl Gun : ammo)«cathfl-egtoa') \CaCautg, €537) . = 3L (Cybcfli C240; ER} '= 5L Quiran \CACsefiZ3,_ 4- W i WWW“ Nut): {MAX =03". . . T .. ' 30: I C Nun —- {Mil-0.6) \CEUR3 i {a mgr-«2.3 4 Ccnfiw EWM'QU' 7. (20 pts) (3.) (10 pts) Write the composite matrix (in the standard basis) of a counterclockwise rotation through 60°, followed by a reflection in the line y = 33. You should not multiply the matrices. Mus“) H (a (pm-5°) also?) me [_.__—._.J , _ 0 L '- £63655me 1 o ‘Cmus’o +mq§ofiflm6° ammo Lsm ng" ways" 0 --| -s—m Li’s" jams" fin 60° com? ’\ \x 0:“ cm soc ant/10° [to 60‘9 whee 0] . Ha qo" - 053cmO 3m 60" 06360" or [.0 l a [053600 -—meo°] r 1 0 blfleoo C6360” - b) (10 pts) Determine if the given transformation is a linear transforma— tion: ' '1"th —9 R2, T(v1,'u2) = (111 +Ug,2v1) wrw = (vi-W. .vuwa) ._ CM =(w‘i. 0%) "-9 (\h'i'wi +W1 ‘i'ui‘i',’ 2Vf+ ZUH') I W. TU) 1TH») : (UH-qu 2V.) +[wt+wzjzw1) _—_ (v(+vz+wmw2,l Z‘vl—e’éwy TLCU) = (.CVl-ECVL 201.) >361“ ETC“) = OCVi-Na'wi) Linear “Ran fimm‘cq ...
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midterm2_practice - Sowwious W 2 NAME SECTION MATH 21-241...

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