06spr-m2sols

06spr-m2sols - 1'.--'(8 poifits) T111?2 ~—> R2 is...

Info iconThis preview shows pages 1–13. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

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

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

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

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

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

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

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

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

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

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11

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

View Full DocumentRight Arrow Icon
Background image of page 12
Background image of page 13
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1'.--'(8 poifits) T111?2 ~—> R2 is the linear transformation defined by first rotating ebunter— a clockwise through an angle of 17/ 2 radians, then projecting onto the y-axis. What is the matrix for T? Le-K— Ring—"'3‘ n21 s-‘mmd‘ co:- rcfi-ai-(‘on 09 We Elana Coun+er,chC[¢t-U{SQ b1 Err-acix'qm, qml K421“ FR'SJK CLAJ “HA8 MG’HI‘K Qat- R (WES was (1‘30 (90% {A (96131 H“ \S'g): 5‘ max: «3.3.: [fl 5% ' -\ Q We Mai-fix E? OX. Nevd- Cm} “‘9 W‘Cfi'fifi C0“ a) («M‘s t‘g Sim:\ar k'o We ‘5 QJCeéi‘): ghee a Is l ’m «A? {grannEs 9 [Ads m+fix g‘mce T: ?°P\ (mafia ‘93“? W159?” H'g ' . . ac; 0—1206 ' Makwa cS [0\1(O ‘0 ‘ (ME; = - 5. = [3]. review ewylm M.s\z a ?(§t\ = #2. :[i5] St‘aeo E1 “5. out We a_-qx}s_ 2- (16 points; 2 pages) 8' (a) Find the inveISe of fikoz‘oo '" °1Ioo 10—300(+12\ 00\'LO{ ao\ \ \ko Q3QO§O_\‘_‘ '00 \ lot 00\.2_O( 302 C": -\t-i 20% Check: Edi-r7. we? ‘0 6, Zoe 20“? O (3! —RO’2. 307, la 0 °" \k——\~L\‘\:O\OL/ 2.0-3 10H 0 at (ioS‘E- Q‘AGCLLt‘wfl owe 0Q We? is emo‘fif,‘ V1.0 be vae N56 v€3k{‘.\ _ 8 (b) Suppose B is the basis of R3 given by kngyfimn . and T I R3 —> R3 has matrix qurbc (Remember 2 B wee-{1.3 @COOTO’AEAQ‘LPSJ A (we MM can» mus mm QQM¥0$QA+§ (Le. gmdaafl Engig-coetdiwqt-egBJ Q fut-As 63-Cmrazm’ces {who Cemescmefifi, wk COMP‘AQA‘GS id‘i'fl g “Cmmit'mfi'fi. _.\ O 1 201 L.‘ (33 2' k K -( ——t \-l - a1\—2 7.0-3 362 -5-0_4_( 3. (18 points; 2 pages) Let Q(:c, y) = 23:2 — 4mg — yz. (a) What is the matrix A for Q? gee P. m are Levquooosky car VHAQ Sena-at SYS‘I'WVL Lt (b) Find the eigenvalues of A. Efflemldues <9 A are We tacks (SC ‘fefie Chqrqa¥e¢~ffifq ?c{~(um {Gd M -.- x“- ' *2 k: Mm ~12 +1 pm __—. Nana-“v: 31-34 = (AHA (1 ~31 ’3 7: “133- 3 (c) What is the definiteness of Q? '3 face A La; ax Pas€HUet and cu waged-Ne e (Seuuqlue/ G) ‘i 5 ‘IVKCflegamt‘w. 4- (8 points) Find the determinant of ' MCMY Vaf‘l ad" 0 “5 [2 «—1 0 3:l on we ‘QOUOWCAj 0 1 _1 0 gokuHcfias 63m. —2 0 4 —2 ' 2 “2 1 2 -po$si‘ol‘e. SG‘QHOIAI 7, —\ a 3 '2. -\ ea '3 O K ""\ C5 -'-_ 6 ‘ “k G #1 O ‘-\"? “1‘ g _\ u‘ | +42; 1. —'L \“L 4,“ O _\ \_l “21 1 —\ 3 ‘-'- \ -( O __ a :: noun-m - -6 o O ‘3 \ o <5 o -l Scab”? {cm I'- EKPfi-wfic‘nfl dath Zm‘ mu): 263 7.44% no 4- i “1 “‘1 _..(_0 -1 0—2 +0 *2“; 1‘11 W '2“:ieoi. ‘4‘“; '7‘“ 41-92. 2'3 :1kl1\¥3\1l\ ei*\1 lk 1&1) 5. (5 points) Sketch four level curves of the function F(-'c,'y) = 3:2 —y- Be sure to indicate the heights. I -_ Me. neeci +0 'K1_%:C ‘Cm- H C \IaKUES ,09 C: (5“ ‘3 X - “gym each (1-: «MO, \,2. C:’\ C20 6. (6 points) Use a linear approximation to estimate f (0.2, —O.1), Where f (my) = 6352”. a: t. *’ b'?(a\ (“K-a I where -—‘ _ is c‘sGSe *6 4. Remember +Me Po'm“? 15 W03? 3 2; [5- W9- elcxce we‘re {Mered'e ‘ a {<5 <1 {ghee m mkere 1+3 Nanak» SC; here §=C&,t&\= (0.1) “OJ‘X aka <1 300$ choke (3o:- 3 lg (0,0\. (AiMos'f‘ may Q‘RAQ!‘ Gimme a“? 5 makes A Qfidfi 8:06“ no QQSEQV «New '1. D4133: 2143* +3 228%]! So. bflcfi: biefqo‘) = [o x] 7. (6 points) Calculate the second—order partial derivatives of f (56, y) = In sinwy) and verify that fmy = fyw. gs S\A(‘L\a\ *‘ ngcos'CK\9\ gm :_ La C05 (in?) + «36.03 (in? " N31 9“ (“‘3\ I. - gma 2:. y; S{y\(_fi\é\ «— KCOS£K~3\ - 28% SM (ray @vflugk— rote Qd‘du Chadd ?Q{€ f 8. (8 points) Let f : R2 —> R2 and g : R3 —+ R2 be differentiable functions such that: . “13):(1’1), Remewx‘oer ‘- 'HAQ Caer‘wcfi-We “(1, 1):“), 4), - “fire is '56le . ' g(—1,0, 2) a (2,3), 6Q a Cameos: I . pm 3)-=-[° ‘3] eccdud- oC— We cilec-LUai-NPS, , 2 1 I ' [ +9c9 where N:- _ “2 ‘1 QQck Q‘JQ VG . ma) 1) _ [1 0], makes Sense. . Dag—1’03) : 3 31} wcqoge j? 0K\\t sees %C'\ROI1\ (a) Find D(fog)(—1,o,2). W/ [ because (9:313 _ see _' I ,J ‘4'- (%L~\,o,2\\ h§('(.©.1\ "..': b?<'2_.3\ b§(i-\{Ol1\ ‘- 2? alga 24131-4 <5 ~31 lix'scsl ‘5q-(. We § oh We {refi- aexisxg 596$ Lk (10) Find D(fof)(2,3). film's” \gecaum We «i an M» \ é rfg‘k‘k' akif Sfies "“ -* Cm) =- (Ffizgfl E 9 {2,3} 9. (10 points) You are climbing on the graph of the function f (m, y) = 21:36?! + '92 + 1. At the point (1, 0, 3), in which direction should you climb to ascend most quickly? - C \czeLg' was?) :2 [New] vg 0,03 = [fl . To Make Z: {30% («crease w‘os‘l- etui‘llc‘Y’ 30 'm We airecham o‘c' . IQ You gS‘QQet‘ *0 SEQCK‘Q c94imc§fcm leA Q unid- Vec’i‘m“ 14: [3 Wu Yfl: f‘qg 10. (5 points) Find an equation for the tangent plane to the Surface given by $3 + 9,3 + 2.3 = 7 at the point (0, —1,2). ’5 '5 "s _F(x,\3,%3= K*3*E 3-? . 13x1 _ V F (W&\ = [367' _ _ 31% O ‘ VF(O’__[)23:: [3} IS Fereem—JCQOIQP \“2. +«: {fie SUVCQCQ oak- (0,“\,2\, 30 ii”; q Aer-Mad \lec‘lmw +0 We flame we mend—- WWW (Z, 2 2— §c> O (x—m +3(‘.OLH\ ’r {262—73 =0 oQ= (9%k +QE'RZO 11. (10 points) Circle True or False for each of the following statements. (a) True A function f : R3 —> R2 has 3 component'functions. CGMQQMA’V Questions 36 92% We m “" in. We case {have are. (b) / False OA\Y Q . Every symmetric matrix has an Orthonormal eigenbasis. ‘Tlnl‘? is, a vesirq‘l-emeal— @Q— We SQQQWl Wearewx in Se C‘lt-Coa '25 (6/ False 0Q LQUQAQQIAY_ It is impossible for a linear transformation T : R7 —> R6 to be one—to—one. "Ware «M Sauce-cal ways +5 See easy. one '6 _W~a Renal:- NdllC‘l-Y Week-em: qukua +“UufiY(/_n=‘:ll chug glance Vanda UA 52' G , w? W\U‘Ss‘l‘ (“We nullc‘i-Y(A\ Bk (d) True /@ go A has a «MK-vivictl “Alsace, s-oT is mt There is a. function f : R2 —> R with continuous second—order partial derivatives ‘ E 2 ' - such that fm(:1:, y) = 2333; and fy(:1:, y) = my . due. In 146: case we would knee guinea: 2X lav“? '93}: 3 (e) Tru / False If 0 is an eigenvalue of an n X n matrix A, then rank(A) S n — 1. Again flew. are Several ways +6 See Wis, afiol ouge is we RawhpNaUH-T “Kareem. O is an 9‘ erman was A”)? =0? =6 gov Some iii-O“ So noUC‘er (155 —>—\ I Gael £0 (souls (in ‘7: unv- vtollt'ltyfll‘l é v’\—l. ...
View Full Document

This note was uploaded on 01/12/2010 for the course MATH 51 at Stanford.

Page1 / 13

06spr-m2sols - 1'.--'(8 poifits) T111?2 ~—&amp;gt; R2 is...

This preview shows document pages 1 - 13. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online