\ MATH 220 FINAL EXAMINATION IDecember 13, 20011 Name ~v\ u.. ~c "'oS ID Section #___ There are ??multiple choice questions. Each problem is worth 5 points. Four possible answers are given for each problem, only one of which is correct. When you solve a problem, note the letter next to 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). THE USE OF CALCULATORS DURING THE EXAMINATION IS FORBIDDEN. CHECK THE EXAMINATION BOOKLET BEFORE YOU START. THERE SHOULD BE ?? PROBLEMS ON ?? PAGES (INCLUDING THIS ONE).l MATH 220 FINAL EXAMINATION PAGE 2 1. What is ~umLiUli of following system of equations @ Xl b) Xl c) Xl d) Xl Xl + 2X2 + X3 1 -2XI + X2 -2X3 3 -Xl + 3X2 - X3 4? = -1-X3,X2 = l,x3 is free. -'1 \ -:l ~ 1'0.) Q IS {) ~l' L , \~ rk 1 1 J = 1 + X2,X2 is free,x3 =-1. _\ -0 -\ 4 tV S 0 5 = -X2 + X3, and X2, X3 are free. [~~ I -IJ= 2X2 X3, and X2) X3 are free. N ()Q)o jrJ \) \[' 2 Q \ J o a {) 0 o 0 Cl 0 "X.\-;-)'l-e:.-= -1 )l\ -=-_\_x~\€q\.\C\-b..QV\s: "><'2 1 ~c\u.\::{OA ?L"2.. = -\ I\J -::.. 0 A 'B \ ~ ~f'<,.ll.. 2. Which of the following matrices is in echelon form (but not necessarily reduced echelon form)? 1 o 32]a) b) ~ll @ [~ ~ i] 1 0 d) 0 2 [O~- -MATH 220 FINAL EXAMINATION PAGE 3 3. What geometric figure mformed from the span of the vectors m'H] ,[i!] ? ./ ./ .J . ~ ~ ~ a) a pomt. I ~ --'" ,'s ::;?aV\ ~l 'f\ ) -'" \l'3} \('3' ~ :£~ OJJ CoL A I.U~b) a line. A L~ ¥-2;;/3 1. \fJ e... V'LI-d. *,0 d,..t.k;1'h ~'-'L(£) a plane. d,VVI. u,\A \ ,. e.. Y()..M-\~ A ) u.I~ck ;~ 9~ tod) all of 1R3. ~ V'cc...tV"> bed"' 4 ?\'\lot-CQ\U ~'" G A . rJ 0 r.J\)\\ ~ l \ 0 Q_\\ \\ J lffi'0 (j)5 ~ 0-\1A~ l ¥, -;i->. .~\) = [ ~ -\ 5 \~ «) -\C( 'q IQ () 0 O-olcA.Y"'\Vl1'"' SO ~VVI ~IA ::: ~-rhu'-<'-(A~ two pl'vor I As "-N 5 u It> 5"f' "'" 1iii iV; -' v:; '[ is ;;L -d.<~ .-" ; C ~0.9-.;.. e. EY' h ..... ·1 4. Which of the following sets of vectors span 1R3? c= w], m, m, [mA=W], m, [!]} a) A only we ¥\Led. 3 L:V\.L.t.I..IrI:l-I'n~~Vectot":r . b) B only S 0 4'~t-clJ-t3 CtLN'-WO+· $ r(M.-tQ:3 c) A and B @ A and C :3 f' ;Vo t:> ) SO A 5pClM1" tR3 , 0 G-~rvr Ii; [1~ ~ Q 31lQ)~tV ~~ o 0 Ll_ <:) 0 db '3 0Ijo-\:~/'\ J \a) \ \ \ ~3 0 o~ \ \ \ 0 o N 0 CD -'1. -1-50 C 5pvvn~I r,:L 3(,v C; I , .:2.. {) 0 0) ')._ If< '3,0 ':l. '-0 vl1 \\ MATH 220 FINAL EXAMINATION PAGE 4 5. For what value(s) of h is the system with augmented matrix G~ =~] consistent? G) It is consistent for all real h. is consistent only if h 8. c) It is consistent if h 8. d) It is never consistent. [; 4 -'3] [/ 31N LIh -6 0 h-K 0 ~ Uf; 1/ no .J-),j0 tr1 cd; I::Vy-Ulha -t4 ;~V\L J h IS-" J l;/-J-f~M [0 o ; ] .0.. Yb )0 Lk ~5~;" Go~;s~ ~oV' 0 aJ\ \fa..\lJ,..(,Y-~. k 6. Suppose A is a matrix with three rows. Three of the following statements are equivalent. Which statement is not equivalent to the others? a) The columns of A span jR3. A has exactly 3 pivots. (c,ve, Jr;n, t /c.no vi ~/ fh~;r &-0 IU.VVl Vl')@ A has a pivot in each column. d) Ax = b has a solution for all b E jR3. ,~ot"(' Y'I ~ ~u,;,,, cJ e·JanJ c ~ ~ f).-I ~ r~~ 43 . };;v..... A ( lCvr) 0 ......\ MATH 220 FINAL EXAMINATION PAGE 5 7. If the augmented matrix of a system of linear equations has the reduced echelon form shown below 0 CD 2 0 0 -1] 0000)03,[OOOO([) 2 then what is the set of solutions of the system? 1 0 a) x 3 +xa 2 0 -2 0 1 0 0 b) x~ n] +x, m 0 -1 @ x= I 0 3 2 0 -1 d) x = I 3 2 0 +Xl +xa 1 0 0 0 0 0 1 2 0 0 +xa ?t-I) 1--:=. ":. t~ ~ ~S -. bCA..s: c?t~>x..\.\ ')\..). + 2 ')(.~ = - \ 'Az..t -= 3 ~S = ;t.. ?t\ -x.~ I'X'20 0 1 0 0 ?t':i "X.y '"iI.? <> -\ -t ?t\0 '3 ;2 \ ~, ::: -\-')..)(. -:. . "X..c, :, ;;2.. 0 \_°2C Hg ~ 0 0\ MATH 220 FINAL EXAMINATION PAGE 6 8. Find all values of h such that the set S=W], m, nn is linearly dependent. ,t , I-VI '1'.3,1(" a) 0 UJe. \"'\.£ (!. d /:;Q h~J Val ues-J "----" ~(WI XI VI + ?\. '2.. \(l... -t x'3\1"'"3 ~ 0 ha-rfOr wh,'c h ~ c) 2 501Ckt-;Ob'\ ( t:k___ t-v"l () 5" +-k (kin -In'II;c..J 3 a-r Ilw! O/VL... ~f\-V-V~·04 b-) ~ 1 -J 1 -/ OJ -'> ~ ~J \ 1 -1 I -'J 0-I h'i"2 0l \fl 2"2-"3 ~ ::::-. 1 h ~] ~ [: 0 J ~U 1 _/ h1'2 01 0 :2 -I 3 '0 ,I It == I~"~·",hl(.. .J -'~ ?t,:1 )'> ~ 1 -1 01(j) -3 0~[~ o A.-I 0 9. If T : JR2 ---+ }R2 is the linear transformation which first refiects points in the xraxis and rotates points in the counterclockwise direction through 7r/2 radians, then what is the standard matrix of T? If) ~ .....\~e\ ~ a) 1 I ~ = l ~J elb) I ~ -11 ~ = l~j€..";l. ~-ez' @ c) [6 [~ 6] ~] + -"" ~ ~ ? /'J' -e'2-Sea Vld evrci f"Y' CA-./;-y:~ A<; [7(e:;'j 1i.;)J ~\_~ ~l\ MATH 220 FINAL EXAMINATION PAGE 7 lO. If T is the linear transformation defined by the formula [123]T(x) = Ax, where A = 4 2 2 ' then which of the following statements is true? a) T is one-to-one and onto b) T is one-to--one, but it is not onto @ T is not one-to--one, but it is onto d) T is neither one-to--one nor onto A= [1 ;) 31 N rW:l '3 1 ~ ;). J J L() (9 \01 o ",--tolk ee..<-k YOW ~-:y T ,'s--dL1'-l-i,.--"" (JI'VC-/ () t\..L-;.. -to _0 ~.T /'5 n-o-l-AA=O)1..3 .$ ~ tree "tAn bIt rOY ==? 11. If [1 -1 1] A o 1-1 , 001 then what is the second row of A-I? ® [0 1 1] b) [0 0 1] c) [-1 1 0] d) [1 -1 0] 0 0 -1 , 1 0 cl \ 1 -I N 0 -1 <0 0 1 …