exam2_key - Math 304 Test ll / Name: \f/ Section: Problem...

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Unformatted text preview: Math 304 Test ll / Name: \f/ Section: Problem Full Score October 25, 2010 Your Score o In order to get credit, you must show all work. 0 Read all problems first. Try working on them from easiest to hardest. o No calculators of any kind! No cell phones! 1. Circle “True” or “False” for each of the following problems. Circle “True” only if the statement is always true. No explanations necessary. D (a) m False et X = {V1,vz,V3,V4,v5}, where v,- 6 M. Then X is a linearly indepen set. I) (b) False 1 Any set of vectors in IR” consisting of a single vector is a linearly inde— pendent set. (c) True False If A is a noninvertible n X n matrix, then Nul(A) : (d @ False If A is an invertible n X n matrix, then Col(A) = R". (e False Subsets of linearly independent sets are necessarily linearly independent. (f False If A is an n X n matrix, whose columns span R”, then A is invertible. ‘A . . a b + 1 . . (g) True _ False The mapping T : R2 ~+ 1R2 given by T b = a IS a linear transformation. (h ‘ False The set S =‘{ (i Any set containing the zero vector is linearly dependent. or + 2b = 0, a,b E R} is a subspace of R2. ~4 , lax} be a collection of vectors in W. {L24 l 2 . —1 ~2 2. LetX= 1 , 1 _1 0 .. M a) Show that X is linearl; dependent by finding a linear dependence relation for X. *l‘ QZ—‘f‘t “012-? ll“! (0’). [1 1417043 5‘35??? 353 -l Oil—2, 01’6 oo o 000 C3: l" C‘::2I (IE—:3 (. —-> ‘1E'.l+3ltl+ a] fig] b) Name a vector that can be removed from the set X so that the remaining two Vectors have the same span as X. (Explain!) e (0.) -‘t .'1_ [22:1 5 ‘ 1D 05ml) 3' q =2. ‘l 3 ' o (3' ’UI M. . ‘il L o .. a (re +M Ire—c.4047 in X, Ha, veg/w be re- Pita—0’ QM. Mm- .9" +lWlw'llfi-Q o—Izlau'lwo waits—D, 1 (3) Let V(t) = [t1] . Determine all values for t such that v(t) E Span(X) and exhibit 1 one value of t such that v(t) g2 Span(X (Show workl). ‘ _ ' "' 9.!“ ‘12,“ Will [OE—I 11-1134 .7 0030 .9 0-, :t-l —7 01H. It 0*lgt'l O! ( colt \ \ “I ' “Wch l# t=O: é X). \ ‘ l 1 1 3. Let v1 = [2] and v2 = Given that (v1, V2) is an ordered basis of R2. a) Write el and eg, the standard basis vectors in W, as linear combinations of v1 and v2. Llé‘gé 3551-7 L3} £32.? ‘0}? [237.5323 1’ star as], [?]= {sip-[Ab / 0 b) Suppose that T is a linear transformation from 1R2 to R3, which satisfies T(v1) = 2 3 1 and T(V2) : [I] . Use the results of part (a to evaluate T el and T(e2). T(‘é'.)'= Will-MEI} 3l§l‘2 "" if: , T(é;l = —-T( [5.1) + T([$])= 531+W= 2‘; 0) Write down the standard matrix of T. 2. l T: Rafi/4?, A = Elle.) 7752)]= l? l (f... 1-1 i —1 0 . . 4 0]) lsaba81sofR. 1 oo-o/u-Q o.) M b) w. M w {\ic’w'ow, he. 546‘.) 3 ) -1 , HX:\/ m X: 93 r] ' ~ 0 *l 01 c) Let u E R4 and K (11) = [:2], the coordinates of u with respect to the basis B. 3 04 Determine the coordinates of u with respect to the standard basis. Km = 3;]. leo M.K(a)=W'Wo/a c3 \ WM Wd’ MOMM "1 ()0 “H (OE/’49 HS—f ll 03 C' CI‘CV 0‘ l 0 '5 Cpl-C2. =71 HNO 1 2 l 5. Let M = 2 4 4 l —l ~2 0 l . a) Complete the following: Nul(M) is a subspace of 10+ & WO—v‘lu) Col(M) is a subspace of (0-3“ ‘lhfl. (4) b) Determine Nul(M) and a basis for Nul(M). —-.7.t0 “(2(0' (7.10 “120-! zq?2,_.>ooz.2.-700 fiLoo «4-10, 00!! 00 [52311313 —- “3 o 0.00 X3.qu 18] me {my «Fella? iii). c) Are the columns of M linzg'rly independent? If not, find a linear dependence relatidn among them. 0‘? ‘11)ch m [R3, WOLlO—FC 01W ,6” clawehM‘l'. _/414 “firm lien-car WM NM; \ "l '2. I o “o 6» 1L2\1-VZI+ 0H+ o 3,] z L0 J .o o ‘ l5 ...
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exam2_key - Math 304 Test ll / Name: \f/ Section: Problem...

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