Winter 2007 - Hall's Class - Practice Exam 1

Winter 2007 - Hall's Class - Practice Exam 1 - 1. (a) (7...

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Unformatted text preview: 1. (a) (7 points) Define what it means to say that the set {171, 272, . . . ,flp} of vectors is a“linearly independent set”. (b) (6 points) Define the “span” of a set {171,172,...,17p} of vectors. m 8loll/l" OF x:\,“‘)\)/pfis 73 M W 0‘?— ’3 MD Q4VULO/V Cr/V‘V‘bimok’hmj at CNN”? . ~ g 2 .. MQ. Sfang‘v\).,.lvp7)~ix‘€‘+-.-+X’F§P\ XCEYK’IS —' (c) (6 points) For What values of p is it possible to find vectors 171, 172, . . . ,vp in R4 so that 171,172, . . . ,fi'p are linearly independent? Explain. 0 \ “96.4,313 kg) (3‘)» . 0:2,; sutka 0+ 3- is & LIE. b l o I 0 / \ ) 9° W‘L Cam 41nd L-‘SI. Saks .9 wk“ P=\'2| 514' \g' P7”? W W .Susslfizvh y\v‘§..+xfsf :3) wow h II E (unuowws a 0W\~1 4' “va5, 81: {A— mus’r \MMM jaw: vex/MM 4 Mo. “0‘4ka sMhmsIWSiwe mam“? \va-fiefl L..D, “il- ‘3’ 4 - (d) (6 points) For what values of p is it possible to find vectors 771,172, . . . , 17,, in R4 so that 171,172, . . . ,6p span 1R4? Explain. W 3 lot as “lam/4. % S 590.445 {2.4 W 4’“ WWW d{' S and OM VQGW l“ #80 MM: \2“)$0 Wan/U3 PzDg we (M he; Suw Q (WM/wt \N K (Wu)? 0" {*4 Veda/Vb “Mi 4448 30‘ W“ 390/94le‘ 1.0 w “WA-\- M W ctl- Land- A VW) emfpm w-Q hwvi ‘7 ...’ w PM 46... M x5) vm-ix 3 =33 UAW—1 Hol- oduocwbo ‘ l; .vr .? . . \ ‘ r\ e W‘“ M W 1 \nwrg 0‘ “win-m‘ Swan, 44M. Mal-rut \ Page Osflw 0&M9V‘ \KWGM WA \ Sc‘l‘”:fs wnmfi’r *6“ . 2. Suppose A is a matrix whose row echelon form is I * * * 0 I * * 0 O * O 0 I 0 0 0 0 0 0 coco- 0 0 0 where I is any nonzero number and * is any number. (a) (7 points) Does the matrix equation A5 = 6 have any nontrivial solutions? Explain. No- We W m (Baez VMWQ‘ $‘V‘Q §;;; 35 \/ 6?“€‘- W9 Eve-L. ox sthh‘“ VCL thefi (b) (6 points) Are the columns of A linearly independent? Explain. jg). W KS 0x .?|Nd+ \m WW3 CWX Let T : R4 ——+ R7 be the linear transformation given by T(f) 2 A55. (c) (6‘points) Is T one—to—one? Explain. Ly... .+ ’rmrng) m A? #43 So Amqu Bw‘r W“ lob ® wt CIA/M— m“ )K‘CSZS) so Home ’Wfil =T(\3) 751:3: , 80 T TS awash“ (d) (6 points) Is T onto? Explain. ‘7‘ Q ) NO- 1’ rszm’m 69*; sets 3mm" 5k. szso 3. Let T1 : R2 —> R2 be the linear transformation that reflects vectors through the line 332 = x1, and let T2 : R2 -—+ R2 be the horiztontal shear defined by T2(é'1) = é'1 and T2032) = €2 —— 261. (a) (5 points) Find the standard matrix for T1. . —3 -> - O 1‘ SkaWls 734 § Q; J 5° 1‘15 a,“ < ‘\ Z 22 ’( \3 i \ C) T1(fi\\ : it 7’ TL()Q:,\ : 31,22: 2 V :. ‘ \ H; :(mm thzb\;(‘ ’25 ( ( o x (c) (5 points) Find the standard matrix for the transformation T3 de- fined by = W T5573 “fut-nae) =9) U33 : [BELTS :3 YT51:(\ "2M0 \ A ’74 0\ \o\’<‘\o\ (d) (5 points) Compute the image of the vector :3 = (3, —1) under T3. um > (4. \X3\ .-.- (“13* “W: (@B x o * \~3+‘o-’\ 3 4. Suppose an economy has three sectors, Agriculture (A), Energy (E), and Manufacturing Sector A sells 60 % of its output to E and retains the rest. Sector E sells 10 % of its output to A, 40 % to M, and retains the rest. Sector M sells 20 ‘70 of its output to A, 60 % to E, and retains the rest. (a) (7 points) Construct the exchange table for this economy. (b) (6 points) Denote the rices (i.e., dllar values) of the total an- nual outputs of the Agriculture, Energy, and Manufacturing sectors by 19/1, 193, and pM, respectively. Convert the table you wrote in part (a) into a homogeneous system of linear equations in the variables pA, mg, and pM. (DATA. +O’\P€ +O'L PM: PA- ? ~O.LeFA_ +O-\PE+O.2?M 1: 8 0’” PA- hb's PE +0va = 3)) (c) (6 points) Find a parametrization of the solution set to the system you wrote in part ('33: °" “‘L ~0-b 0-! 0-L ,a «>44 0-\ 0.7,. -049 0 0,4 0 “is 0'“ \N 0 ‘04 0-? 0 “0‘1 0.? N o —-o.\ 0.; a) o 0.4 43.1 o 0 a o o 0 PM 5 Sven _0, ~ _ Ffi’fiiPM' 23"?!“ a: PA» 04""; F6 'ZPM P Pg 2 FM 2 l’M \ (d) (6 points) Find a set of equilibrium prices for the economy, in which all prices are positive. w “PM = we, aao . 00 “WW” 1’», = 62(2ijlele PE > 900,000.00 Page 4 ...
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Winter 2007 - Hall's Class - Practice Exam 1 - 1. (a) (7...

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