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Math 205B (^) Exam I page 1 (^) 10/09/2009 Name (^) W1F;lS!r,( rPNI7;III ~r/
- Let T: RB -,>Rb be the linear transformation defined as T(x) = Ax where A =
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: 11
lA. What are a and b? a = QJ b = [j]
lB. f1nd T( [~}.
Tai]): AIi] ~I'U}"[;]+ '-In=I:f]
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!C. Let b ~ l~J' Find 8J1yjallconditions that bl, b" b" and b, must satisfy;n OM'" fu, b to be;n the 'ange 01 T.
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V:> 0 , 3/'1 () () -11/2.0 !> (j <> 0 0 I (J 0 D I " 3j}- -2./>" (J () 0 0' ¥s - 1/S"
'Ii. fJ'ft". f t~$1~ IwuuJ/ t CrAfljj"i
~
f
0 =^^10 I^ + IS1L^ b 3 - 1./I!> h I.f
0 0 =: L2. + % 103 - % h'
lD. Verily that s ~ l{~J saUs/iesthe oondiUo*) in 10.
Doe~
{
O.i 0 + }'ID - %-./S- = J.2 - 2'3 : ~-(, = D /
O? S +}~:/D - %'I$"= S + 8.'2. - t'3 = 5"+1'-21 = D /
IE. Find all x such that T(x) = s.
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d,.~ "rref-liI;' j f!1/ s] r,.,Jce1 g g g -f] ) "'" 7(,,):, t {-- >
X, -= 1'2. -+ /"i X,
X-z..-::. - y2. - Y't X:I
XJ ~ fee.
/'Jot. .;k.t Ie tR~ k p~tI .(/~l. ;/ s
lIT rvw SCMJ' X, -)4 >1 =:X ~l ...{~'" 5(
l~"'" :i'~(S )(1.-+J4 'lCJ= ~~obl + ~~Cf
~ -%; 10 -+g..' IS"
:= -~ + J
(Ii ~ Mf£ It> =:-~
I ~ /rt:f ttlf- o%V - '!9~/;'" !,
Math 205B (^) Exam I page 2 10/09/
Name .f(/ftJ,dr,l ./, fdJ2~.$
NOTE: This is pr
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m 1 continued!
IF. Supposed ~ f. Usethe condit"''""in (IC) to findallvaluesof doand d, forwhichd i, ;n the rangeof T. (Note you will be setting up a little linear system, and you should use our linear algebra techniques to solve it).
~ h.[WL {O:: 10, .. ~ hJ - %h't
I D : b..' %~ - ~L.
~ i C r;J a J11~t sM' ~ 11($, Itt b) w<- rt7fl'ff.
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0 '" -3 + t dl -~J'i 0 = 2 -I"%~ -J'sd~
7L. MJAJJ ,.",.fr1¥1,(,ff~ !Iui ~ih' ~ r* -7} 1
1
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r-' [
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L% -* -l. 0 I 3D =7{)'1 ~sO
tG. Is T onto Rb? Explain why or why not.
No ~ 4 iW ks /() /Rb k J"r) T(~)=J: L n.. JD/~~
Suet. d~J .(#/11:: # ~ fnl»~f ~ sal;} ~ ~ e?;/,r IA fc)
tW ckvj, "", /;: -Ct s..Jf :t......
tH. Is T one-to-one? Explain why or why not.
No; L f~ / /1 " Y/Jn:.a.h ilx <'/; /11'- 1M. 7(1) = l: c;;.. Jv.., /1'<>" 4 d><.r,/..L.: ;;:;
/YJ~ vllAl ~ cP -mlfJ f C .f'/ftfih;t ~/~s I~ (Ie).
J(J¥;s/r,I .s;/vhOlI~ Math 205B Exam I page 4 10/09/2009 Name J.F
- A model for an economy uses 4 sectors A, B, C, D. S~r B consumes 10% of the output of A, ~ce that much of its own output, and 2/5 of D's outpu-t0sector C uses 1/10 of its o~~utpu@nd t~rema.inder is consumed in equal portions~
by the other three sectors. Sector A consumes half of D's outpu£Und vice versli!YD also uses 1/2 of B's output but A an~ D consume none of their own output. Sectot C consumesas much of B's output as B itself does. '. (}) 3A. Remembering that the entries of each column sum to one, what is the exchange table for this economy? 0 It- B CD
0 j ~o - _J'DIO_ 12hi ~A
~O 0. ~-1 1 ;/0 ~ I ;lsa I ~ B
Yr ~ Yio to I 4 ~
IL _I l; 1j, 0 ~ D /L(D/ /L.fl..") I/o(i.
Jr-Ii 1e1f(f1 (~J @) I~ t /J£~ ;. ~ ~~ 1~{r;1 Iv ~ k()~ Il #i
('~fhetlcot!t. t/11!tk/~ t'~~ ~. )
dtkr hI/Il~,( fa) ;I.. l(eJVit'/ll> j','fl"~ ()1Ie II "= )
3B. Suppose sector D has an equilibrium price of $179 billion. What are the other three equilibrium prices PA, PE and Pc? Label your answers.
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O~ + 10 fs + % Pc -r ~PD =/A
%0 ~ + }Pa + 1;0 Pc -r JFb=: 11
~ ~ + Ysfp ~!Io Pc + Jofo = Pc.
Ji ~ -<X~ +!;of: -r ofb =~
tt/fi; JtfrzJ;~cJ ~ (^) I ~/
~ fPO ~ / .P ( r'P Jitt. / 1/-td hJf1fJ
(trt~JId!J / l a.;-*.t }WJ(lJx ~
PB Pc PD
-)
I (^) '
X
I 0 0 - mjtr:t =- nYffI PD
to /Iv
Jc} (^) - rS- 7s-1:
Po ,.-J 0 ( 0 7S0/1'1'1 /SDf1'1 Po
0 0 I /tS / 191 /1s}f'1 fp X;
-}Iv )0 0 J
'24- (^) 000 (;J f, ' D (p i"U.
X
10 -I 0 l = IJ'!
- vt4jJ IW r: I t /)k! iii Ps ISO^ 6l/;d^ I. 'JI D, Ec = lIS
~ath 205B Exam I page 5 10/09/2009 Nmne~etfd SDltillJn.&
- Suppose T: Ra -+ RZ is a transformation. Give the definitions of each of the following: 4a. T is a linear transformation.
- /R~ /1« ,;, ,;. //lJr,{v' T/I¥IS£"'J/;"
{::=9 (j) 7( t1 + V) ~ T{ t-) + T(V) IJr a, ~ ~ t ~ a.-
(j) T (! ti ) ~ s T(ii) k, et, ii € /!??i a./ s (; j!
4b. T is onto RZ.
&Ji ~ T h tJf"fr;f( r~ J~ tij tGI!?2 ;L.~d ~t- X<fj('"
f;r l/~l~ T{ x) :::!f: ,
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[
X2X3 + X
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4c. Suppose T : R3 -+ R4 is defined by T .:~ = 3Xl+ ~2 + X3. Show by example that T is not a linear
X3 2X2 + 7
transformation and that it actually fails both part."3of the definitionin (4a); use all-differentnumbers as entries in any vectors
-- td-frr tL'[i J aJ9=[f] tr W<"fk. .'
(j) TlUTV)=T(U].[lJ)' I([[j)~[<;~;:]=[:o
. ~ T{~) +T(&) = T(UJ). r([1]) = UJ+[:]J=f;jJ
~ [{f]f [;il u;' S~tM~! -r ~ (¥Vi (j)j It tljA,A;' h 'I~, ~~VL.
@ f$il ~ Ii) a.J soz.
T{2~) = T(2[!P' Tan)= ~;~~ = r;n
~ 2 T~) = 2 r(LI})
= 2 [
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