Midterm%20Solutions%20Spring%202009

Midterm%20Solutions%20Spring%202009 - Midterm Spring 2009...

Info icon This preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Midterm Spring 2009 — ECE 174 Put your NAME and ID number on every page of this exam. There are THREE (3) QUESTIONS on this exam — be sure to answer every question! Name: %C>iufi om ID Number: Problem Points 1 / 20 2 / 4O 3 / 40 Total / 100 ECE 174 Midterm — Spring 2009 Name and ID: 2 l. (20 points) Subsets and Subspaces (a) Determine which of the following subsets of R” are in fact subspaces of R” (n>2). Prove your answer (i) {Xlxz-ZO} (ii){x|ij:O} (iii){ x | x,- = 3m} (b) Determine which of the following subsets of R” x n are in fact subspaces of R” x ”. Prove your answer. (i) Symmetric matrices (ii) Triangular matrices (iii)Diagonal matrices (iV)All matrices such that the determinant(A) = 1 0C3 :W WORK QM“ HA6 Sube’l‘ lo be. ocfiumpace) H’ ng‘l— lO-C CloBéol Undéf‘ adohhon (Am anvil Y\Au\b3\‘\mfi0‘(\ (MU ‘ . WW1: &06 mol’ SQHS Ml fimce i? WQ MMWj \m3 0k: «l var/$0 omi— o? “WK §€$l’, N53:— ot vedfbf SubeGC‘e ii? /\\/\3<;1 30618885 Pm anal M! fimcé ‘rP 2X130 “linen éléfi'Hfi‘A: 2X'k4' 2L3, =0 M ioOQ : o< :A’w _’Th'ns us a \Iedof‘ Subspace, it?) films Sabscwi Al am M1 .83yoflr wa v\-a| X’/ EX\)3X\fiXUM 7?) XS (H V\"l { ’Man X’l '3; 3% )gxx’a‘fll 3‘“ l?) 2943 ‘j' : flx‘mb ,HWMO ) ) (and (XIT‘jg Omd (XX; €04)“ )ocrsx) M ot3h_\x$:io<x\) 5(000 ”,5 (“xiii ) W\% S. a wed’br SULBSTPQCe . ECE 174 Midterm — Spring 2009 Name and ID: 3 b} '3 Mar Par W. “(NV “*0 bfi 0K WOWW \ {H andr M1 mufi’ be Kym/60‘ AOHF’HQA'S Oré'lmrj WWWfiSL adolHfiom) and ECala‘r Muffi?(\(‘a'fi07\ '13 OFdfimr (demCVvan/URQ MMfi-iyl'um'fion b3 m (flfl Rum/WK , U TWS XS Q U6 Abr Subwacfi :3: a“ : 0‘11 and I“ ’U/flg "3 mg avedbr 3MBSF066, 1+ 0‘06: no’f' Qafifiqs (M %r A uwer manjutlar owk B {bwef , “Yflarvsudaré A ““75 33 d€fl§£ , “\M’Ms I; 0L Uédlf RESPECQ- “AOMLH‘IOLA 030 OMOflOV‘OJ 'W\OKJW 1% 0M Mulh?\\1<&‘fim \OU\ 6Q COWQ—‘WW (\fi‘éflefljfi dhaokORQW‘E/S . ”\T‘mz \3 Viol— a VQdDF $0003?th RN A VWV‘ > (AfirCm :\ ) ”(hm defied Pv\:o<“o\e*m\ _ ho ECE 174 Midterm — Spring 2009 Name and ID: 4 2. (40 points) Let the m X 11 matrix A, AsXHY, be a linear mapping between two complex Hilbert spaces X and Y with inner-product weighting matrices given by Q and Wrespectively. (a) Derive the form of the adjoint of A, from the fundamental definition of the adjoint. (b) Consider the inverse problem y = Ax. (i) Derive the algebraic condition for a least-squares solution to exist. (ii) Derive the algebraic condition for a least-squares solution to be the minimum norm solution. (c) Derive the pseudoinverse of A for the following two conditions (i) A has filll column rank. (ii) A has filll row rank. (d) Construct the pseudoinverse of A for 9, W, and A given by . . 923.58 0 0 0 0 0 0 3.45 1—2] 0.02—0.01] 0 1 0 0 0 1 0 9: 1+2] 9.76 2+] ,W: ,A: . . 0 0 1 0 1 0 0 0.02+0.01] 2—] 513.29 0 0 0 1 0 0 1 Where j = fl . Give the pseudoinverse in its simplest possible form. We Roland/vs 11) pews (a) \ (em amok (a can be \‘Bund 'm ”PM lame and msolufimg 17> Mmfwork ‘7‘. For “far‘l (d) ,bemuifi The matrix has a“ Column rank )‘l’l/YC ?S€udo‘1AV€FS€ ‘13 index/(Wad 69‘ i7. and NAB ”(he {Eryn <3e< homework assfinmerfi’ Ur anal flb’l’E W+ A \% Ffml> Ar: (/19 wA)"‘A“w :(ATWAY‘ AW. 11“— 13 alxo QOSHtj oLQ—le(‘m“,\_(o( “fina+ Wfilfl (eiflwotlem’lg ATWZAT‘) (mot A'VA: I) (50 “W3“ A+:A\— ECE 174 Midterm — Spring 2009 Name and ID: 7 3. (40 points) Let two DC currents 11 and 12 each pass through an inductor L1 and L2, L1 7E L2, respectively followed by a summing junction from which flows a designated target current]. 1L12. Recall that the energy stored in an inductor is given byE : 2 It; Li (a) Using a purely geometric approach (i.e. do not take derivates) determine the values of 11 and 12 which minimize the total energy stored in the two inductors while attaining the specified target current]. (b) For the special case of two identical inductors, L = L1 = L2, use your answer determined above to compute the optimal (minimum) value of the stored energy and compare to the value of energy stored in a single inductor L through which the target current flows. By what factor has the stored energy been reduced by using an optimized two inductor circuit rather than a one inductor circuit? QVOM KCL \IUC. MUQ, I‘VE-£2, 7.: )L/S\€,\Al fl3Ax) A’Zii l] 3x:[\:\ T “’2 \mHA A 0M0. The 'lb’tal enerokj 1% (3616A bfl L ”Z Z “L ‘ 1 O :L i 4* I :LT lelxn - LLl' 2L1; ZXJL )& 01.; Therfire) we @ro‘olem “1% do «QM-line Mianum norm S‘Dlm‘l’lm it‘l’lne SL§+€M 3: AX finch 'fi— '13 QXOC'HLj (3‘1“?qu k or. some. town as ”the Feslsrof 'Pralolem 9:)VQA 'm +146 homew (3113+ replacf/eaclo R; m er leWMWK Wm“ ewe wrresfonélvj L10? TM exam flVWSHoA, N30 (\ofie M ‘HAQQQM «Problem 13 awaklkj s'umrfler as H— kq'i q M0~ohmeh§onoq hnpud" SPICE) Mid/WI ”(’wa +bree.)_/W\{ Solufion Kg * L1 T T — M ”P __ " ,_ j “‘2.” " ECE 174 Midterm — Spring 2009 Name and ID: . ,_ _: ‘03 13A TN? 00x86)l\«12« Z330 “Wad” F, ‘L \ 1 ”L [ l E;—}Z L<i§>JViLLfiZA :‘fILI WW3 qvfiwu’ ‘\’D +1A£ °_>\V\cQ\t,'\Y\duC+OF “Nd C onwarlk 3A9: LIZ) wtsee ’(koi’r “HM Shredemrguj 140$ Sn—efoxds) been reddcefi Lab} mQde‘ 0‘92. ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern