THE USE OF A CALCULATOR, CELL PHONE, OR ANY OTHER ELEC- TRONIC DEVICE IS NOT PERMITTED DURING THIS EXAMINATION.

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1 MATH 220 NAME So\,t\\OV\ '. FINAL EXAM 18, 2007\ FORMA STUDENT NUMBER INSTRUCTOR SECTION NUMBER This examination will be machine processed by the University Testing Service. Use only a number 2 pencil on your answer sheet. On your answer sheet, identify your name, this course (MATH 220) and the date. Code and blacken the corresponding circles on your answer sheet for your student I.D. number and the class section number. Code in your test form. There are 25 multiple choice questions each worth six points. For each problem, four possible answers are given, only one of which is correct. You should solve the problem, note the letter of the answer that you wish to give and blacken the corresponding space on the answer sheet. Mark only one choice; darken the circle completely (you should not be able to see the letter after you have darkened the circle). Check frequently to be sure the problem number on the test sheet is the same as the problem number of the answer sheet. THE USE OF A CALCULATOR, CELL PHONE, OR ANY OTHER ELEC- TRONIC DEVICE IS NOT PERMITTED DURING THIS EXAMINATION. CHECK THE EXAMINATION BOOKLET BEFORE YOU START. THERE SHOULD BE 25 PROBLEMS ON 14 PAGES (INCLUDING THIS ONE).

2 MATH 220 FINAL EXAM, FORM A PAGE 2 1. Find all solutions of the following system of linear equations: Xl + 2X2 X3 = 1 2XI + 5X2 2X3 = 0 5Xl + 12x2 5X3 1 a) Xl = 5, X2 = -2, X3 l. Xl = 5+X3, X2 = -2, X3 is free. c) Xl 1 + X3, X2 = 1, X3 is free. There are no solutions. -\ \ J l1 '2 () 1 J (\D oq) n'2. -1 Q -1 N S -2 (;) "-' Q 1 tl \ \ 0 '2 o -y \ I ")(.:.:1. bcas \ Co 'X.':?;.: bov\... 'X\ E:. +"X..'3 -= - "2..?t.3 ''iii ] 2. Suppose is the augmented matrix of a linear system. For which [ -5 5 h 3 value(s) of h is this system consistent? a) h = 12 h =I -12 c) h 1 l1 :2 -\ \ ').. '3 -\ \ -\ 1 -\J 2 E \ -\ tv <> \ f'..l 0 -'5 o \1+\2 \ \"5 hi-\s -'2. l\) d) h =I 12-2 I \. - S h 'lis \e "'" 3 0 0tl\l'Jti It\:. "'-"t' \:1. '* C 'i..,e. "':f

3 MATH 220 FINAL EXAM, FORM A PAGE 3 1 1!], 3. If A = -2 3 and b = [t], then describe the set of all b such that Ax = b [ 6 8 has a solution. a) b 3 = b 1 + b 2 b) b 3 = 2b 1 + 2b 2 c) b 3 = 3b 1 + 3b For any bi, b 2, b 3. -'" l111 LA 61 = -2 0 \ b q b, J '\ 1 1 b\ 1 1 '\ b \ ] r-l Q 0 b'2."'"i'':lb\ t...i Q '5 '3 'b-a _Cl 'l -'.1. b:=, - f,b\ _ 'C,-\ - 3b\ b \ rj <:l 1 _\ ':1..\0 \ '\) (:, -t -:2. b\ - \: - '3 b, Q \Q-(iY b ;. \0::, _ '3\0 \-\ N led'q '\GJ 1-\ \ is b\ l f\ \...o.s 0... \l\0,>\- '''''' -E"o-cM V<>I.AJ 1 2 3] 4. If A = 2 5 5,then what is the row of the inverse matrix A-I? [ 124 a) [ 1 1-1] [6-2 [-1 0 1] d) A-I does not exist. 2 'l. ::;, 0 0] 1 0 OJ o "'\ rj \ 1 '\ '1. o 0- 'V -"3 1 1 \ -I -'2 \ tc N 5 5 o10 \o -"1 '\ o 1 [1\ "1. 'i Q 0 1 l 10 () \ -\ I r.j 1 o 0 \'\:) -2-5! \ Q 1 t:.l._9.-.jjj tfvu.--<- ',.!. t.l.. t'cf,oo. '..Q... O,)I:\c...,.'""," \co\:"\o"'"

4 MATH 220 FINAL EXAM, FORM A PAGE 4 5. If A = [; and the linear transformation is given by T(x) = Ax, which of the following vectors is in the range of T? ').. \] rj I \ '2. \\ a) 2. Lt () '- <> 0-1..J '2- h] N r' '2. ri 1 b) [ 1-\ Vi. l',:, 'I:l -Vi [ 4 ';)..0 J L 0 0 c "l.. \ \ f\ '). \1 f' d) [ ] \ 'l. l.\ -I tv l '0 'l.j2 1 ('J f \ 'l. /21 v'" ihcif: Tt ;S;/1 +Iv<., e.- 4 \" -'".... "k A u-. \;:;:' Co "',!!:"S, 6. IfT([ [;], and T([ = [ -i], then what is T([ i]) =? a) wt'ik L 1} ihl 4 n A.",J \. C\ l : 1 -t C2 1-= [11 b) [ ] '0 l\ NI o \ c) [ [ Ii ] C\ :::: l.\ <::l =:; \ 1"( l11) "1 ( \ 1-+ L,1) = ( C1J;- \ C. ti] )

5 MATH 220 FINAL EXAM, FORM A PAGE 5 7. What is a parametric form of the solutions to the following linear system? Xl + X2 2xa 0-3XI 4X2 + 7xa 0-2XI 3X2 + 5xa Xl = t, X2 = t, Xa = t 1-2 OJ 11 1 OJ N \ \ -3 -l.\ \ b) Xl = t + 28, X2 = 8, Xa t [: \ Xl = 2t, X2 = t, Xa = 0 o IV <:. \ -\ '0 o ' I AI "" "Il:o I -I?l '3 ;5 Xl = t, X2 = 2t, Xa = t Nr: "'-?\. == )\3 - i?if -<- t o.-v1 XL 't". 8. vectors VI, V2 Va be given by v, =, V2= UJ V3 and n] The basis B = {VI, V2, Va} is not orthogonal. What is the value of X2 if the vector y = is given by the linear combination y VIXI + V2X2 + VaXa :2 -\ I {(.;ow 'ri/.duu- 'i-'l. \f!-s b) 5 l: c) 0 '2 -\ l j l1 r-j '0 2 \ I L\-2- d) -3 Lj -'2 '00 'l. 0 '2 \0-1..\ r-j [ =U -I 1 J j "?J "- 0 0 tv C C \ c-2 5 0;:) \ '0 0 "A.:=, III = '=> "A.':l=-'l. -= 0

6 MATH 220 FINAL EXAM, FORM A PAGE 6 9. Let the vectors Vl, V2 and V3 be given by and Vl = l/tj' V2 = [ m The basis B {VI, V" V3} is orthonormal. What is the value of c, if the vector y m is given by the linear combination a) 1 Y = VlCl + V2C2 + V3C3' = -"'" \rl _ \1[2 1_ b) -/ //2 d) T : ]R2 ---t ]R2 first reflects points through the horizontal (Xl) axis, and then rotates clockwise by 90 0 Find the standard matrix of T. I\?l'l. a) _ ] -e'; = b) [ 0 1] r:- \..e,p-e2. - J [ 0-1] ")\2, c) 1 0 -e; _E\= TO -']..;1.. PI=- (l'(el) Tce;.)j= \_\ 0

7 MATH 220 FINAL EXAM, FORM A PAGE Suppose A is an n x n matrix given by A = [al... an], and the linear transformation x Ax is onto JRn. Which of the following are true? Ctal Cn-lan-t =an for some Cl,...,Cn-I in IR. (1=)-. :x.. l\ awlq.\.v,..u..v\\ II- det A =1= 0 / tr): '\s \V\'le,...h6\e. l TAl) III - The solution set of Ax = 0 is a two-dimensional subspace of JRn. (F):. \:) 'vi.g,) u,...\'d- IV - The linear transformation x A-IX is onto JRn../ I" 11.. \. s.<;>\lll-}'v'"!-\, <; Ocls IJ ;"'\1 e-,\-\ a) I and IV ;t)1" II and IV c) II and III d) I and III ] 12. The matrix A given by A is row equivalent to the matrix [ Which of the following is a basis for the null space of A? 3] [2]} { A\ -=- -=1/-;...)\-::, l" "4 a) 2, 4 1l'). =. _s/l.\ A 3 - V<..?\y {[ -3 2 "'4:;'.5 8]} \ So -1 -x b) {[ = - 7/2] [ 4]} "4 c) {[-, ", {,? (-r2 1'1 is.. bc-<-s c's -ttjy' }I«I It,

8 MATH 220 FINAL EXAM, FORM A PAGES ] 13. Let A = Any solution of the equation Av 0 may be written as [ where t, S E JR a) 1, d) o 2 v = I 1 t + 0 IS, o 1 o 0. What is the rank of the matrix A? rk'y1l hul Ii 2 # 4 Co\I.l"'-""'., f\ A -t d i""" NeAl A -=. S = #....,- {-\ 'Yeu"J,::: f\ ;- '2. 5? {\ -=- '3 14. What is det B if B is given by a) 2 b) 6 c) 24 [ 1 2 3] B = ? Y 0 ((':1. -'> R'l. -t 1<.., R3, ---'!I 'R.. -+ (J IOrr 'rf'i'"bi 1\.. dekts 6) =

9 MATH 220 FINAL EXAM, FORM A PAGE Suppose the linear transformation T : IR3 I--l- IR3 is given by x I--l- Ax. Further, suppose T reflects any vector in IR3 across a plane through the origin. What is the dimension of the - 't) eigenspace associated with the eigenvalue A = I? a) 0, d) 3 AVl't F;.v:;: ' d """\- VV'Y::>""'- Uk..J.. If!N' t "'--- II Q.c\-'-OA 6"\- -ie. I<"'\l"""") =--.f"" \" l.::;;-) A 'S-e va.- A\j! -=-- -;:- toy a1\ q;)\a t:-"'--- "Vt.. T h.,'s,.., -t"1a At; ).,= 1;s Ok J<- \ o...j \I-4.:lor 0", tt-v- e. -e.: e-. Gy" '} bs- >.. i / <1 / i-s tw(j 16. Suppose det A a and det B = b. What is the value of det ((3AT)2 B- 1 ). a) 9a 2 b CorN.e&- C)V\ -t.\.""- b) 3a 2 jb c) 9a 2 jb W"'ict'l ttl"{. v)ot- c..( _ d) 3a 2 b A o.",j t o..""'l 'v"ulv\ dq1-( (3r:P\:l. ) = de.\.\(3f\l/') \1 ('2 v'\ ').. -= \det-a)' -= \:>

10 MATH 220 FINAL EXAM, FORM A PAGE 10 5/2-1/2 0] 17. Let the matrix A be given by -1/2 5/2 0. Which of the following is an eigenvector [ 001 of A? m a) b) [ -1] d) 1 r 1 II] 6cc.Jt:kr -; l I *A : (:j = 1 \ l:j \.5;,. ->2 oj r-'] f-:,] f-\] c-: -L! =\i./ [ % OJ i-i] l-0'2j :;;,., 0 0 -= y- -::f. ')... 'l. 0 Q \) 1 II I I 18. Let the matrix A, given by A = be diagonalizable. That is, A may be written as A = P-1DP, where D = 1 0 0]' P= I 1-1] 1 2' and p- 1 = [2 1 1] 1. What is the matrix product PA6? [ 1-1] a) ] [ : x X X 10 6 ] [ X ] d) [-56 2 X 10 6 Ab= plobp PAb = f' r- l Db p =- -r eft> -::. Dbp 'Of, 01\ \ -\"\ = [ c IObJ L-l '). J b b - T 5 rs "\ L-\Q6 \0'J

11 MATH 220 FINAL EXAM, FORM A PAGE 11 1a 0] 19. Find the characteristic polynomial of A = a [ a 3 a a) (1 -.\)(2 -.\)(3 -.\) b) (1 -.\)(2 (1 -.\)(.\2-2,\ - 3) d) (1-.\)(.\2+2,\+1) r-" () 0 o _) -It!. ) -= 0!2 -A '\ I -::: (1-'),) \ '} _ \ = lv).1 ( l>--i.)l-;") - <» = (1 -i\) ( )?- :2.') - '3) 20. Suppose the matrix A is given by. Which of the following is an eigenvalue/eigenvector pair? (Note that in the choices below, i = J=I.) [-i] a).\ = y'2 +.y'2 de+.(,a -"AI)= C) \ 1-), 1 '\\ = () 2 2 2' 1 -\ \-A b).\=il+iil r-i+ =0 2 2' C9.\ = 1 + i, 2.. d).\ = 1 + i, i + 1 1] A= 11"1.. A_;2'1:. ",' - = \-j:1- [ -i i. 0-\ (A-')l -6} = 1. oj EqU6.t.on.. to 1Jv.. VI),j\!-: _?I.I - i.,. 2. -=;.(... --x..\ = -.. x. '2. \ -""l ""- L is C>J\... A.'Jed:oy.

12 MATH 220 FINAL EXAM, FORM A PAGE What is the distance from y = [ -:] to the line through u = [ ]? a) y'4 A -'-" (]J3 'A= -- -"" y'2 -""" -"'" L'2 ]...:l1li. U- _, u.. = '2.,... =- d) y'1 """ - '::L 0 u-v. :l w -t"'-sl. -k.wt. 'S \\ \\ _ \ -= -I '2)':l+(_1 - c)':l +(1- r:l.) '). =-[3 22. For which value of h are the vectors U, = [ and U2 = [ orthogonal?,h=l --"'" h=2 L4' U2 0 c) h = 3?-(1) -t 2\-::L) -t' 1. = 0 d) h = 4 ':2. -'4 -t -= 0 '2. -t' -=- a

13 MATH 220 FINAL EXAM, FORM A PAGE Which one of the following bases is orthogonal? { 2 } ccjl t;n.- w{ coy:) r1j 7 U"L h-or'l-\ a) 4/vT8, 0, 1/ ,. "L _ ' \ 1/vT8 1V2-2/3 '3.1.,... 1'& '3ffl b),, [j]} -'" U\ U3 - _h";::. "" r-:;;- - =-- -=tc ---"" -"- \h.u'3 -= () -t" '2. -'0 -= :2.. -:::p 0 \1'l..'\1.'3. -=- -\ -to..,.o =-,"*C W],n],[m -- -=- --::.. '-l\. \l'l -=- U."l... U3. = U \ \J..2:, = [-1/0], } 1/V Find the distance from the point [ ] to the plane spanned by ] and [ _ ]. a) c) J5 d) 5 '\ '3 '-A. u\ -- '-l\ -t -- U r: \.U ::.., ll' \.l2,. --- ll",u \ -->0. - "-\ t.. \l-z. \\ \\ \\ -;:: -/ \- 5 -t (2-- -=- -flf 2

14 MATH 220 FINAL EXAM, FORM A PAGE Construct an orthogonal basis for Span{x"x2}' where X, = [ ] and X2 = [ -: 1 a) [n and [ -! b) [ :] and [ n and [ ] d) [n and [ ] \f,:t+1 = \(j _ V; -= 1. 'l.. - 1\_,,-_' "" if" \"1 ""' -r- = ;; "\ [-i] \ -- J\ \-3\ l 1,"2 f = ( 1 ' l Jj 'y"",- dn\. I:;)\J'C..Q. ("" , Let the matrix A be given by A = 1 5 [o 0 corresponding to the eigenvalue A = 41 \ \ L a) la -1.\1. 01 =- \ o () 1 d) 3..,.-k e;8(?/n sro.cjl..., What is the dimension of the eigenspace oj"" 'O-oQC 1 \10::::' o (:) 0 coo vo r f -t:t:l Null 4 A -1.\ I.. 1'v'-U- 'Siv..ce.. ')":; 4 \'S"' <1\""" e.n5 '" l...-\-.q.. t..a... \s -