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# exam3_key - Math 304 ‘ ~—— ‘ Test III Name 1 g i 2...

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Unformatted text preview: Math 304 ‘ ~——. _ ‘ Test III November 30, 2010 Name: 1 g i: 2 Section: 1 2 Full Score 20 20 Your Score o In order to get any credit, you must show all work. 0 Read all the 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. (a) q-- A vector space has a ﬁnite spanning set if and only if it has a ﬁnite basis. ‘ (b ( Let V and W be vector spaces with dimV = n and dimW = m, respectively. If : " a W is a linear transformation which is onto, then n 2 m and dim (ker L) = n — m. a linear co oination of vectors in S in only one way. (g)“- The dimension of a vector space V is the number of elements in any basis of V. (h) [email protected] Let A and B be n X n matrices. Then det(AB) =det(BA) if and only if det(A+B) = det(B + A). (i) [email protected] Let A be an n x n matrix with det(A) = 0. Then A has a zero—row. det(kA) = k” det(A) for any n X n matrix A. l E i l E g E E 2 2. Let 1193 be the vector space of polynomials with real coefﬁcients of degree at most3. Let S = {V1, v2, v3} be asubset ONE; with v1 = 1+ac+x3, V2 = —ac—a:2+a:3, v3 = 1+x+zc2. (a) Show S 1s a linearly independent set. (ﬁf X: (41 X;XZ/ XS) bQ mil; 1"“ g MCI KX Le“ c/de‘I/é .l‘mV‘S—lbﬂm mm 3 i5 lKW MAL? m Pr» 43> KX<S> \5 (£54 HAW W\ ' K l a; l wall 5:351. “MM (23:2 5'0 mt e #otcilsx _. l "‘ 'M‘“? , w ‘ .__,..:=, A“ (9% i ‘Tgy‘t‘rii'*';@--<r Vseg.-g Do “I '90 KX(S> ;@O( “”13 “SA A} l l O 0 l" O O G If» lime/Ma Méaefle/wI/‘JC Heme ‘5 a: (MA {mt/4);): (b) dim(lP’3) = gig and dim(Span(S)) = 3) . (0) Determine a vector v 6 P3 but v §Z Span(S). M 9161 MWV_V "“30"" ll Gl 0‘ l l ll 5 ”4...? 0 it; SQ Kirngl/S; O"l O meX C? {H C Qal‘cé’l/t 0 "Vii C E O “l l C, lEO cl VOEMi d'ﬂ Doc, A‘Wl'l’c z” . [A "a as: 0mg vie/show [El with CLIoIICifO is WI")T \W 5%“ CKX 63% :0“ POCVl—€CU&(CLV/ [%1¢SWWCKx<S>> :3.) l1: (L591) -: x3 ¢ 3%” (73> (d) With the help of a), b) and c) determine a basis of 1P3 which contains the set 8. Explain! ‘30 \(mcw {vxclé pee/\AQI/xce exl‘em atom lQ mime/I “”3: Q0 ”MA CC“) ﬂex :5ch Sb "Why limeXi‘WIXZIXiH'II—f;X33 KS Lih‘EClYl a ihggfk4wd‘2gx (9% We, each basis {a “P3 has 4; eleI/Iw/“S/ \$0 St}; >53 {3 m lac—Isis in» a @IIﬁIa/WISS (some eveJ limearl {hdng mks/Wt :sejc with if elemﬂts “4 P3) is; m Sam‘s 5%U’ E) 3. Given P3 and P2, the vector spaces of polynomials of degree at most 3 and 2, respectively, and the linear transformation T : 1P3 —-> 1% given as T(f(a:)) = f’(:r) + f”(:1:) where f(x) 6 P3. Let f’(x) and f”(:r) be the ﬁrst and second derivative of f (33) (a) Let X: (1, a3, :52 ,rr3) be an ordered basis of lP’g and Y = (1, :1}, x2) be an ordered basis of P2. Find the matrix yTX. KL 73>le j, 131% (TC45>\L(T1L))\K/(T(r)] 2W7) 2 13 2.123% (7) ’\ Z <3 , (o)l<(1)ll<K(Zx+2_ \K ”5,2262%: 02: 2 2 , [KY KY Y >K O C 0 \$ #2922me (b) With the help of a) show that T is onto. ‘\ T is Oh+0 (:27:> YTX elgﬂmg (pi/5L1) \W/‘QLU/ WW? @17V0MLYE (“N Y2)! is m RE? @lveadj 323 Vamb\$xr3 Wﬂél “173022; (c) With the help of a) ﬁnd a basis for the kernel of T. Is T one—to—one? Agra L22"? L‘QC mu m. [/22 W ”a V2. C2" {Zr/ 0 i 2. a Q~~axg——5r <0 (0 O X' :3" O 0 a 6 5 (7 o l 5 --*~;> 3 i Marcy ._ .2 ~/‘ 0 L: .000 ,5, V1”? 9/2. O O l.- V,~er{—~2vé g0 C? 3:2 O (36 o€bL£70W 5 . v? EDP/£2. M" v 79, 0 W m6; (hi) is 0: Lam‘s ”lbw NbllT lx mml Ci> \‘2 0\ bags; ”W” {582 l, o T ‘3 “01 om i’wome as KerT LL03 Lfetjmwd ’“ Pa ((1) Determine a basis for P2 which is contained in #T(X ). “T (X): l o 1 aw—Z 33x12 éxj 712 ﬁrst VQQ‘lYDv Q E TC x) l5 0C ww+r\w(;kl Umr’x’ti/ Cambimmalﬁ‘o m '0; Cf‘féxu/ «@1322; gm T09 60.. b 3 Van/x Liv 'rejaervm/dio w 112. 94’) th L) g: Q ”5’6“”; m<l 105) 9.03) 5",?qu ( VCX>> m T<gﬁ 36‘“ [X B2: ‘ 2 5 TCX} {0) 2 14,222+st +ex'5 is a spawning 22+ a P2 Com sis-L23 2L1? :52 2122 E, 2 (was; T1223 1‘ T(X)” \$03 25 r, W {L bﬁ‘i/ll7 WV aim/Mala i; CONlTLLl/xacl in TWO ' “MWWWWWWWMxmﬁmmmmwamwaww E i i L E i E [ E i E 4. Consider R2 with ordered bases [9‘ 222:: (01,02), the standard basis, and X z": (wl, W2) with W1 2 [_ﬂ and W2 2 [Ff] (a) Find the change of basis matrix 1.;1 X that changes from coordil’iates with respect to X in R2 to coordinates with respect to E. Jr [acmwwjﬂ “1] as a; an a; cure ﬂax/eh in ”terms; 0? Eggs E‘CQ pl) (b) Find the change of basis matrix XII; that changes from coordinates with respect to E in R2 to coordinates with respect to X. EIEZU . i .1 vi X . [ 1 O' E‘ﬂgg‘ﬁ i ”Z i O VJ‘V‘G E ’2 ”E l O I 0 ~( ! l c 3 ' (0) Let L be the linear transformation from R2 to R2 deﬁned by L(e1) = [—2] and L(e2) = [3;]. Determine the matrix ELE. ELE: [KE (ma) garb-9?]: Kﬂfﬂ . K{ [1:] -.: [f if? V‘) th 'th t‘L. XL:A~‘L-°1‘: () eermie emarixxx 76 XX}: E t E x ﬂwxm‘mammmmmWWQMQMQWWWWQWWWWWWWQWMmmwm-anmmmmwﬂxmﬁmmmwmm:, 1—3: 1 9 1—96 w A: QW— Cr Q >< 6% ~ Ma): Q new) \$9 Qeﬁﬂ): 0 z::~> Xl'i >«r—L/ 5. (a) Let A = [ ]. Determine all m such that det(A) = 0. 1 2 1 (b) Let B = 1—1 —1 2] . Determine det(B). 2 1 -1 - ' " H , ‘\ ‘3' o 6 "b ...
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exam3_key - Math 304 ‘ ~—— ‘ Test III Name 1 g i 2...

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