Winter2005 - Surname ”—1 First Name_— se Print...

Info icon This preview shows pages 1–10. 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
Image of page 7

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

View Full Document Right Arrow Icon
Image of page 8
Image of page 9

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

View Full Document Right Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Surname: ”—1 First Name: %_— se Print Faculty of Mathematics MATH 136 Monday 14 March 2005 Question Mark Winter 2005 Id. No.: fl. University of Waterloo MIDTERM EXAM #2 19:00 — 20:15 Instructions: 1. 2. Only Faculty approved calculators are permitted. The exam has 5 problems. Unless oth— erwise indicated, you must Show all of the work that you did to obtain a solu— tion. . You may use any result from class with— out proof, unless you are being asked to prove this result. Make sure to clearly state any result from class that you are using. . Make sure to put your name and ID number at the top of this page. . If you do not have enough space to fin— ish your solution to a problem, continue it on the back of the page, and indicate that you have done so. . There are 2 blank pages at the end of the exam that can be torn off and used for rough work. This exam has 10 pages in total, including the cover and two blank pages. CIRCLE YOUR SECTION AND INSTRUCTOR BELOW: Sec. Instructor Time 001 K. Giuliani 9:30 002 I. VanderBurgh 9:30 003 R. Levene 10:30 004 K. Giuliani 11:30 005 H. Wei 12:30 006 C. Hewitt 9200 l007 J. Muir 1:30 l A MATH 136, Midterm #2 1 2 [7] 1. SupposeA= [2 1+i 1 1+1 ’B‘[0 2—2‘ 0—1 Calculate 2'A + BT0. _ i l \ ‘ 2. Ln 0 -\ a \t 24 \H 0 “t 4- X\\;\\ ZEXL352‘ L\-—\ ~3 ‘FR 2—\ // —1 2 0 ],andC—l: Page 2 of 10 Name: 3 l + [11;][2 23 931% (+43 ’ ~\ i -\ . 2.» 2 ‘_ + ‘ L; L‘s [[23:32 :Z-Z‘X \io—C? 4::ng “. fl 7. _7_ (2 P.‘ + 84‘ 5"?>‘\ _‘ -'L 2; —2 4—2’ ' ESQ / \ ”3“"7- 2X7- 5‘3 3x2. m 343?) mw‘ ($5 + ._ ,,+ QM MW?) (2003; L2 40% ‘17 :5" 8 ““”‘ Z\ -— ”h ”“2 1” “(8 *‘IR‘ Name- m 004.3 1 .5 L\ :0. 2700 DN R Q WWW. w W was 4.13% \Zyok Mo \ZRWKERQ 29?. p e 2.13%232 wl‘IU :M 10133??? m \m go @0046 QQ.%2\Om m Q 00\ \ _ a A O\OO Z 00 «S 2 m OslowQQ 7.00 m m m oogmowcoio 003% 00 M, Wm M OOfl%QQ\O§ OQ\Q ©0‘O m m o H c m 210 zxoo axes o\oo M 2. \3o0 M m \000 MATH 136, Midterm #2 Page 4 of 10 Name: Hz \3 [3] 3. (a) Let V and W be vector spaces over 15‘, and T : V —-> W a function. Define what it means for T to be a linear transformation. l? T E Q \‘m ‘WOX‘FQ, W \l‘ da‘x‘iims Q soj- Q? “$13 wick mop elem m \1 +0 amis/xmm ‘m \N. H Mugs mus (3w addition a senior mH‘xpfigflfiim ,/ Co Tmfih = TLRB‘VTG’B 18“”??? “iv " Q3) T La (‘0 == J01) a Teimrcmev ab‘xei closed MW 0603‘ E snowy woo“: (b) Define T : P1(R) —> R3 by T(a + bt) = (a + 2b, 2a — b, b). (1P1 (IR) is the set of polynomials in t with real coefficients and of degree less than or equal to 1.) [4] i. Prove that T is a linear transformation. 0) new: um comm m7 a Q‘Q-t Es = C\‘*‘\Ot 9; $2.: "I’V‘flE-f:L ) §\ 5“ fileawaR‘) mm max = Lem-mama‘cfi :. Tcsa=cm+1gflme 2b TM? “(323 = T L©+nQ+ (mgr) : < Ck+/‘/\+20C>"’ED 3 2(ci-HQ— (bi—5) ) have” -" (ammo >QQ‘Q*QW a u b +3» \./: (Q+Zb) ZQ’b' b» + (“+23‘QNV'3 ’35 ‘ :. TEE” *"TC‘S-b ‘ . . Mow madam oddvfion 4;; UN timed ow Seolcxr mufij M l3 = 09$ $€WUR3 ) we eel?- TC "‘3 = T 03+ ebb j? = CC%\+'ZQ&>. ZBQ—e‘ow‘cb = (c (mm , e um)», are» fl?“ MATH 136, Midterm #2 Page 5 of 10 Name' - - \ (b) (continued) Define T : EUR) —> R3 by T(a + bt) = (a + 2b, 2a —— b, b). (1P1 (R) is the set of polynomials in t with real coefficients and of degree less than or equal to 1.) [4] ii. Determine a spanning set for the kernel of T. Is T 1—1? Explain. (Gd-(Zia, ZG’lOi b» 7'" Cox Dub) gnu, WT: i R€P\L®\AR :3“): Q +2\O=O Q=Q 20’b=0 gt. 20=C3 [4] iii. Determine a spanning set for the range of T. Is T onto? Explain. (aflJo, 10"“b, b3 T ‘3 get , => 0\ -\— ‘1 X m J: 05 mm mm reamed C) + b MATH 136, Midterm #2 Page 6 of 10 Name' b [3] 4. (a) Let V be a vector space over F and H a non—empty subset of V. Define what it means for H to be a subspace of V. ‘90? \-\ -\b be. cx eubspm, GbV M‘Qmm'wg Promfi‘wa MS’V be. Ma G) M Ev Mg bum/84¢ \-\ Q0 \‘\ mug? m €30de um «(MEMO . \e \3; "Mil QM 4m» @eQQA Q“) *‘\ we? LsL emw W gemw WWW) (Em-Rem ‘w we a Vex—x W (mega, J MATH 136, Midterm #2 1 Page 7(0f 10 Name'fiig [5% BEZM b'Ek l (b) Let V = BUR). In each part, determine whether or not H is a subspace, of V. Justify your answers. (1% (R) is the set of polynomials in t with real coefficients and of degree less than or equal to 2. p’ is the derivative of the polynomial p.) [3] i H = {p E V l 13(1) + p’(1) =1} 3%“ glib”: 21: o +‘ot +56 => o+b+c+b+2c =\ =5 O-‘(lb’ggqfifi 0‘ "‘QH 7i ‘-‘- Mgr»??? =3 Nx*2\5+’33'2::\ \ WI (3‘ +159 = (Co HQ +Qo-‘r \fit + LU—‘QE—E = (Q‘r ”Wt {hm} + Lox-k} -\— Lb+®+ltm1§§ = @Jr’lbfiag ~\- ow 233+ Ea“) l+ D \9333 0mg»; “lo Waggfir a... _- L r = 2 Vlfiék‘ Q? “9353 =>2 : Ms m % .W’m Wales «whose (grins) to )\. cmgsee afifisfi’m . Subgmu gnu” WW: mir- Qlomx MW' Gddthdvy/ l3] 11- H={p€V|p(0)+p’(1)+p”(2) =0} \ A Q3 5w: R." €: 6:4; WA QH ; a A A A“ Q*’7\)l=’7( Qt®\-\- Ox‘Lh-xd‘tz} + ”(03+ MD} + bx (Q3 = L o -\- o} be mpg“ «25% ‘-\ , . . a " s‘ * . cl Wm ookdifikuw. C21: (filaflcfl + 600x 5(3u L753» / = age) =3 C3 ;. \r\ chard vmm' N3 ii W mm MWK twat aames firm 01;) f) 3C) mum “W “ll/u mom GB suwuvfllo (bus. i’l \3 (‘x subspoircGV/fl /’”‘ maW' W~ MATH 136, Midterm #2 Page 8 of 10 Name: $99!? Eflh'ld 5‘ Ek \ [4] 5. Let V and W be vector spaces over F, and let T : V —> W be a linear transformation. If there exists a b E W is such that T(x) = b has exactly one solution, prove that kerT = {0} (that is, the kernel lof T is the set containing only 0). agew 33L TL/AQT’E‘O \flQfi: Q ‘mlsfi\%LSL %CSQ:B £> Soto. T17)“ =1 has 0 umcbw 9:603 7 AN. fim‘dfifilx - max-WM A, at AR filo} must W rawm ‘é *1le “Q3 VGUQ \ {a 29x03 \fi :1 \OWO . . l db V A RQ‘ES 0\ chst \h 931% YQNQM A =3 smm, As we?— has a p‘wo’v ”m men; Vow ) Afi=b has 0 UthbUJL 95C“ 5:132 21%?) (A43 :5 Slums, A Ms, (HG-g) UNW $056“- ) \"t u)\\\ M W W Wicfl; wt"; lb Ali =23 €5ng cows cm “ml/MP, % \D/C' M W ‘Wfihwc $6613” ‘lb AR=§ 3% m NT ‘3 m sfirtfballll a? 31mm N153, m kart-:35) was. MATH 136, Midterm #2 Page 9 0f 10 Name: THIS PAGE HAS BEEN LEFT BLANK FOR ROUGH WORK MATH 136, Midterm #2 Page 10 of 10 Name: THIS PAGE HAS BEEN LEFT BLANK FOR ROUGH WORK ...
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