351Sp11Sec6mt1solns[1]

351Sp11Sec6mt1solns[1] - Math 351—06 Midterm 1 Name: ”...

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Unformatted text preview: Math 351—06 Midterm 1 Name: ” Instructions: 0 TO RECEIVE CREDIT YOU MUST SHOW ALL YOUR, WORK and WRITE NEATLY. Be sure to explain your answers. 0 The format will be closed book, closed notes and no cell phones, blackberries, calculators or other such devices are allowed While taking this test. 0 There are 8 problems on 5 pages. Solve Problems 1 through 4. Choose 1 problem out of Problems 5—8 and cross out the ones that you didn’t select below. ‘Time Allowed: 50 minutes Total(45pts) 2 Math 351—06 Midterm 17 February 9, 2011. Problem 1. (9 pts) Consider mewmm a: + y —- z = 3 2m+y+z = O x + (a2 — 7)z = (1 § For which values of a does this system have § a unique solution? (b) no solutions? (0) inﬁnitely many solutions? A.a:~3anda=3 A.a=—3anda=3 A.a=—3anda=3 ayé—BandayéB B.a7é—3anda%3 B.a7é——3anda,7é3 ; é .. C.a=3 @a23 C.a=3 i i __ w ' _ _ D.a———-3 D.a——3 aw 3 E.a7é3 E.a#3 E.a753 g l 1 - 1 t 3 I l -» x / a 1 1 -— 1 3 2. l 1 O M) C) “I 3 “*6 “‘5 0 “l 256‘ W: a . “l '2 . QM (1+ i O a J} a 0 ml awe; dug 0 (l5? no SQlu‘hOf‘l wm qzncfgg amcl altar} no Solulbﬂvwm 02:3 0r 42% l 0‘5? b3 lﬂflwlelﬂ “mﬁﬁl SOW’W‘S whim Chqu :: 5:) amd a Jr 3 r: C) } ml‘lﬁéﬂﬂ C}: '3 (or (:1 2“. “3 Okchle 6‘ '33“ ’3 (30 M w W- Ci 2. M 5 tall _ Lia‘xqve 30mm u.>\l\12f\ a 4r 3;, a at “55. Problem 2. ( 9 pts) Suppose that T : R" —> R” is a linear transformation and that there exists a function L : R" —> R” such that L(T(X)) = x and T(L(y)) = y for every x,y 6 R”. Show that L must be a linear transformation. Let *3 New? 49 "as; : “T(L(“G§) Ham} 3:- "T C 1430+ Moi “ring 3‘» :3; ’7“ war if) MSW): L. (T CLCCK‘)+LC3’33>3: le’luv> A __ v A LLTC’WW IX ‘7 ">9 LfLX—W)’ = L(§3+LL©> 433, all 3;: 127“. ® Mi: k ‘T( um) \: “T (n LL33} is) Lita) r: L (Tu not» “ “Harman HQ) :kuﬁ) 43m» axxaemiell 73> L “is \lhfcxri Math 351-06 Midterm 1February 9, 2011 Problem 3. {12 pts) Let T : R3 ——> R3 be a linear transsformation such that W<EJ>=EI Tw1<LaJ>xm w: Find the matrix A of T. ,3}; 2 i l ' I Am “T6330 TED ’TLEQ :3 I 2” , \ x } ' ’5 s__.__1 V H 4 Math 351~06 Midterm 1, February 9, 2011 Problem 4. (9 pts) Compute the orthogonal projection projV(x) onto the plane V, where :IER3 : (31+2m2—2x327}. WW :3 3g unit we emf” \"n m d‘sf‘éfck‘igm (in? L, a.) Pﬂv'Fe/Nciiqﬁ ON“ 4:) ‘(3 5; mm); morma\ we aim ml) "1 ma» W » «A U m. vm—w—w wmﬁ’. W 7: ’1 /+~WW hhhhhh M 11 {:3 l\ «1 mwa wz—v (if :1 a 3 w . L x ( 4 am 21 WW: g) ijV (K s: y: .._. {1 3 M41 *3 ,2 1 9515 V a; or ~--——-———- -1 61 3i in Math 351—06 Midterm lFebruary 9, 2011 Problem 5. (6 pts) The linear transformation W = [3??? .355] x represents a reﬂection about a line L. Find the equation of line L (in the formy : mm). L” ‘7 ‘36 P‘M‘W *0 L- “WA woman. ml : W5 abltv'le “s ~+ V1 V3. h- (/6 L, 3 _. -~—>.. -. \.i\.\/ 3» EV i 3:. 2 § E E: i g i is i s i .3 if i E i E 2 § i i i S i é: E i Problem 6. ( 6 pts) Consider a linear system Ax = b, where A is a 4 X 5 matrix with rank(A) : 3. How many solutions does this system have if (a) 7‘cmk([A : b]) = 4? (b) TdnkﬂA : b]) = 3? CA) no solution. . i A. no solution. B. inﬁnitely many solutions. inﬁnitely many solutions. C. a unique solution. C. a unique solution. D. cannot be determined from the given informa— D. cannot be determined from the given informa— tion. tion. mnWA=b”“D = H» > mm (20:3 'mnkl [A :10] 3 = 3 : mini) \\,L ‘59 stem is Mwﬂ’ﬁig‘tﬁ/v}; \$9346“ lg C‘Qﬁs’ig’iﬁ/it‘ \n/ as or: ‘ 3 as 0-? -mkm') ‘ ‘ s i , :7“ \JAknOan5 H m we soluhcsné, _ 15*.3 :1 W are m»€mﬁ€§3 wavjﬁ sowﬁgns. 6 Math 351—06 Midterm 1, February 9, 2011 Problem 7. ( 6 pts) Fill in the blanks: The linear transformation T(x) = :56] x represents ........ .. combined With a scaling by a factor ......... .. Q) <3 M Si .0) ' r. 14 “‘” “1a A. a rotation, 100. T k .93. 33.. . k. B. a reflection, 100. (CPVQQUJS Lowbwwaxk‘ T i’bﬂﬁorn A wﬁm a scamﬂé , C. an orthogonal projection, 1. D. a reﬂection, 10. k g ‘2 .k, ( ‘0 3 .\$\ (4, u lg @a rotation, 10. 1 § E; t t g s: i 2*; a i i 2:: ,5 § § 2 E 3 5.; i é’ g (z g e o & Problem 8. (6 pts) Let A = [(1) a. Compute A2 = AA, A3 2 AAA = A2A and A4 :4 AAAA = A3A. By using the pattern that emerges, if compute A2011. H; ﬂux] {g ‘11 Air-[Latin 1i {2: ii [; fl L: xi =1; n .n \ n _ w l (A: [ \ X g A .. \ b. Fill in the blank in the following sentence: The power A” represents ............... .. A. a vertical shear. An m] K {T \ “,1 [xi 1 a horizontal shear. X7— 0 l Xz’ C. a rotation. M AX\+Y\X‘2_ ~._ ‘1 D. a reﬂection. X 7- B. an orthogonal projection. ...
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This note was uploaded on 10/17/2011 for the course MA 351 taught by Professor ?? during the Fall '08 term at Purdue University-West Lafayette.

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351Sp11Sec6mt1solns[1] - Math 351—06 Midterm 1 Name: ”...

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