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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’).
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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
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(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 111134 .7 0030 .9 0, :tl —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 35517 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'.)'= WillMEI} 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...
11 i —1
0 . . 4
0]) lsaba81sofR.
1 ooo/uQ 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 CplC2. =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 (03“ ‘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
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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 liencar WM NM;
\ "l '2. I o “o 6»
1L2\1VZI+ 0H+ o 3,] z L0 J
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