7KHRUHPLQ6HFWLRQ VKRZVWKDWLI A LV m# n WKHQWKHWUDQVIRUPDWLRQ [ 7! A[ KDV WKHSURSHUWLHV
A.XC Y/ D AX C AY DQG A.cX/ D cAX
IRUDOO X; Y LQRnDQGDOOVFDODUV c 7KHVHSURSHUWLHV ZULWWHQLQIXQFWLRQQRWDWLRQ LGHQWLI\
WKHPRVWLPSRUWDQWFODVVRIWUDQVIRUPDWLRQVLQOLQHDUDOJHEUD
% & ' * / * 5 * 0 / $ WUDQVIRUPDWLRQRUPDSSLQJ T LV OLQHDU LI
L T .XC Y/ D T .X/ C T .Y/ IRUDOO X; Y LQWKHGRPDLQRI T LL T .cX/D cT .X/ IRUDOOVFDODUV c DQGDOO X LQWKHGRPDLQRI T
(YHU\PDWUL[WUDQVIRUPDWLRQLVDOLQHDUWUDQVIRUPDWLRQ ,PSRUWDQWH[DPSOHVRI OLQHDUWUDQVIRUPDWLRQVWKDWDUHQRWPDWUL[WUDQVIRUPDWLRQVZLOOEHGLVFXVVHGLQ&KDSWHUV
DQG
$)"15&3 /LQHDU(TXDWLRQVLQ/LQHDU$OJHEUD
/LQHDUWUDQVIRUPDWLRQV SUHVHUYHWKHRSHUDWLRQVRIYHFWRUDGGLWLRQDQGVFDODUPXO
WLSOLFDWLRQ 3URSHUW\LVD\VWKDWWKHUHVXOW T .XC Y/ RIÀUVWDGGLQJ X DQG Y LQ RnDQG WKHQDSSO\LQJ T LVWKHVDPHDVÀUVWDSSO\LQJ T WR X DQGWR Y DQGWKHQDGGLQJ T .X/ DQG T .Y/ LQRm 7KHVHWZRSURSHUWLHVOHDGHDVLO\WRWKHIROORZLQJXVHIXOIDFWV
,I T LVDOLQHDUWUDQVIRUPDWLRQ WKHQ
T ./D
DQG
T .cXC dY/ D cT .X/ C dT .Y/
IRUDOOYHFWRUV X Y LQWKHGRPDLQRI T DQGDOOVFDODUV c; d
3URSHUW\IROORZVIURPFRQGLWLRQLLLQWKHGHÀQLWLRQ EHFDXVH T ./D T .0X/ D 0T .X/D 3URSHUW\UHTXLUHVERWKLDQGLL
T .cXC dY/ D T .cX/ C T .dY/ D cT .X/ C dT .Y/
2EVHUYHWKDW LIDWUDQVIRUPDWLRQVDWLVÀHV IRUDOO X Y DQG c; d LWPXVWEHOLQHDU
6HW cD d D 1 IRUSUHVHUYDWLRQRIDGGLWLRQ DQGVHW d D 0 IRUSUHVHUYDWLRQRIVFDODU PXOWLSOLFDWLRQ 5HSHDWHGDSSOLFDWLRQRISURGXFHVDXVHIXOJHQHUDOL]DWLRQ
T .c1Y1C ! ! ! C cpYp/D c1T .Y1/C ! ! ! C cpT .Yp/ ,QHQJLQHHULQJDQGSK\VLFV LVUHIHUUHGWRDVD VXSHUSRVLWLRQSULQFLSOH 7KLQN RI Y1; : : : ;YpDVVLJQDOVWKDWJRLQWRDV\VWHPDQG T .Y1/; : : : ; T .Yp/DVWKHUHVSRQVHVRI WKDWV\VWHPWRWKHVLJQDOV 7KHV\VWHPVDWLVÀHVWKHVXSHUSRVLWLRQSULQFLSOHLIZKHQHYHU DQLQSXWLVH[SUHVVHGDVDOLQHDUFRPELQDWLRQRIVXFKVLJQDOV WKHV\VWHP·VUHVSRQVHLV WKHVDPH OLQHDUFRPELQDWLRQRIWKHUHVSRQVHVWRWKHLQGLYLGXDOVLJQDOV :HZLOOUHWXUQ WRWKLVLGHDLQ&KDSWHU
&9".1-& *LYHQDVFDODU r GHÀQH T W R2! R2E\ T .[/D r[ T LVFDOOHGD FRQWUDFWLRQ ZKHQ 0' r ' 1 DQGD GLODWLRQ ZKHQ r > 1 /HW r D 3 DQGVKRZWKDW T LVDOLQHDUWUDQVIRUPDWLRQ
40-65*0/ /HW X Y EHLQR2DQGOHW c; d EHVFDODUV 7KHQ T .cXC dY/ D 3.cX C dY/ 'HÀQLWLRQRI T
D 3cX C 3dY D c.3X/ C d.3Y/
)
9HFWRUDULWKPHWLF D cT .X/ C dT .Y/
7KXV T LVDOLQHDUWUDQVIRUPDWLRQEHFDXVHLWVDWLVÀHV 6HH)LJ
T(u)
x1 x2
x1 T u
x2
'*(63& $ GLODWLRQWUDQVIRUPDWLRQ
,QWURGXFWLRQWR/LQHDU7UDQVIRUPDWLRQV
&9".1-& 'HÀQHDOLQHDUWUDQVIRUPDWLRQ T W R2! R2E\
T .[/D
#0 "1
1 0
$#x1
x2
$ D
#"x2
x1
$ )LQGWKHLPDJHVXQGHU T RI XD
#4 1
$
YD
#2 3
$
DQG XC Y D
#6 4
$
40-65*0/
T .X/D
#0 "1
1 0
$#4 1
$ D
#"1 4
$
; T .Y/D
#0 "1
1 0
$#2 3
$ D
#"3 2
$
;
T .XC Y/ D
#0 "1
1 0
$#6 4
$ D
#"4 6
$
1RWHWKDW T .XC Y/ LVREYLRXVO\HTXDOWR T .X/ C T .Y/ ,WDSSHDUVIURP)LJ WKDW T URWDWHV X Y DQG XC Y FRXQWHUFORFNZLVHDERXWWKHRULJLQWKURXJK 90ı ,QIDFW T WUDQVIRUPVWKHHQWLUHSDUDOOHORJUDPGHWHUPLQHGE\ X DQG Y LQWRWKHRQHGHWHUPLQHGE\
T .X/ DQG T .Y/ 6HH([HUFLVH
T
x1 x2
v
u T(u + v)
T(u)
T(v)
u + v
'*(63& $ URWDWLRQWUDQVIRUPDWLRQ
7KHÀQDOH[DPSOHLVQRWJHRPHWULFDO LQVWHDG LWVKRZVKRZDOLQHDUPDSSLQJFDQ WUDQVIRUPRQHW\SHRIGDWDLQWRDQRWKHU
&9".1-& $ FRPSDQ\PDQXIDFWXUHVWZRSURGXFWV % DQG&8VLQJGDWDIURP ([DPSOHLQ6HFWLRQ ZHFRQVWUXFWD´XQLWFRVWµPDWUL[ U D Œ E F ! ZKRVH FROXPQVGHVFULEHWKH´FRVWVSHUGROODURIRXWSXWµIRUWKHSURGXFWV
U D
3URGXFW
% &
2
4:45 :40 :25 :35 :15 :15
3
5 0DWHULDOV/DERU 2YHUKHDG
/HW [D .x1; x2/EHD´SURGXFWLRQµYHFWRU FRUUHVSRQGLQJWR x1GROODUVRISURGXFW% DQG x2GROODUVRISURGXFW&DQGGHÀQH T W R2! R3E\
T .[/D U [ D x1
2 4:45
:25 :15
3 5C x2
2 4:40
:30 :15
3 5D
2
47RWDOFRVWRIPDWHULDOV 7RWDOFRVWRIODERU 7RWDOFRVWRIRYHUKHDG
3 5
7KHPDSSLQJ T WUDQVIRUPVDOLVWRISURGXFWLRQTXDQWLWLHVPHDVXUHGLQGROODUVLQWRD OLVWRIWRWDOFRVWV 7KHOLQHDULW\RIWKLVPDSSLQJLVUHÁHFWHGLQWZRZD\V
,ISURGXFWLRQLVLQFUHDVHGE\DIDFWRURI VD\ IURP [ WR 4[ WKHQWKHFRVWVZLOO LQFUHDVHE\WKHVDPHIDFWRU IURP T .[/ WR 4T .[/
$)"15&3 /LQHDU(TXDWLRQVLQ/LQHDU$OJHEUD
,I [ DQG \ DUHSURGXFWLRQYHFWRUV WKHQWKHWRWDOFRVWYHFWRUDVVRFLDWHGZLWKWKH FRPELQHG SURGXFWLRQ [C \ LVSUHFLVHO\WKHVXPRIWKHFRVWYHFWRUV T .[/ DQG T .\/
13"$5*$& 130#-&.4
6XSSRVH T W R5 ! R2DQG T .[/D A[ IRUVRPHPDWUL[ A DQGIRUHDFK [ LQ R5 +RZ PDQ\URZVDQGFROXPQVGRHV A KDYH"
/HW AD
#1 0 0 "1
$
*LYHDJHRPHWULFGHVFULSWLRQRIWKHWUDQVIRUPDWLRQ [7! A[
7KHOLQHVHJPHQWIURP WRDYHFWRU X LVWKHVHWRISRLQWVRIWKHIRUP tX ZKHUH 0' t ' 1 6KRZWKDWDOLQHDUWUDQVIRUPDWLRQ T PDSVWKLVVHJPHQWLQWRWKHVHJPHQW EHWZHHQ DQG T .X/
&9&3$*4&4
/HW AD
#2 0
0 2
$
DQGGHÀQH T W R2! R2E\ T .[/D A[
)LQGWKHLPDJHVXQGHU T RI XD
# 1
"3
$ DQG YD
#a b
$
/HW AD 2 4
1
3 0 0
0 13 0
0 0 13
3 5 X D
2
4 3
6
"9 3
5 DQG Y D 2 4a
b c
3 5
'HÀQH T W R3! R3E\ T .[/D A[ )LQG T .X/ DQG T .Y/
,Q([HUFLVHV² ZLWK T GHÀQHGE\ T .[/D A[ ÀQGDYHFWRU [ ZKRVHLPDJHXQGHU T LV E DQGGHWHUPLQHZKHWKHU [ LVXQLTXH
AD 2
4 1 0 "3
"3 1 6 2 "2 "1
3 5 E D
2 4"2
3
"1 3 5
AD 2
41 "2 3 0 1 "3 2 "5 6 3 5 E D
2 4"6
"4
"5 3 5
AD
# 1 "5 "7
"3 7 5
$
ED
#"2
"2
$
AD 2 66 4
1 "3 2 3 "8 8
0 1 2
1 0 8
3 77 5 E D
2 66 4
1 6 3 10
3 77 5
/HW A EHD 6# 5 PDWUL[ :KDWPXVW a DQG b EHLQRUGHUWR GHÀQH T W Ra! RbE\ T .[/D A["
+RZPDQ\URZVDQGFROXPQVPXVWDPDWUL[ A KDYHLQRUGHU WRGHÀQHDPDSSLQJIURPR5LQWRR7E\WKHUXOH T .[/D A["
)RU([HUFLVHVDQG ÀQGDOO [ LQR4WKDWDUHPDSSHGLQWRWKH ]HURYHFWRUE\WKHWUDQVIRUPDWLRQ [7! A[ IRUWKHJLYHQPDWUL[ A
AD 2
41 "3 5 "5 0 1 "3 5 2 "4 4 "4 3 5
AD 2 66 4
3 2 10 "6
1 0 2 "4
0 1 2 3
1 4 10 8
3 77 5
/HW ED 2 4"1
1 0 3
5 DQGOHW A EHWKHPDWUL[LQ([HUFLVH ,V E LQWKHUDQJHRIWKHOLQHDUWUDQVIRUPDWLRQ [7! A[" :K\RU ZK\QRW"
/HW ED 2 66 4
"1 3
"1 4
3 77
5 DQGOHW A EHWKHPDWUL[LQ([HUFLVH ,V E LQWKHUDQJHRIWKHOLQHDUWUDQVIRUPDWLRQ [7! A[" :K\RU ZK\QRW"
,Q([HUFLVHV² XVHDUHFWDQJXODUFRRUGLQDWHV\VWHPWRSORW XD
#5 2
$
YD
#"2 4
$
DQGWKHLULPDJHVXQGHUWKHJLYHQWUDQVIRU
PDWLRQ T 0DNHDVHSDUDWHDQGUHDVRQDEO\ODUJHVNHWFKIRUHDFK H[HUFLVH 'HVFULEHJHRPHWULFDOO\ZKDW T GRHVWRHDFKYHFWRU [ LQR2
T .[/D
#"1 0 0 "1
$#x1
x2
$
T .[/D
#2 0
0 2
$#x1
x2
$
T .[/D
#0 1
1 0
$#x1
x2
$
T .[/D
#0 0
0 2
$#x1
x2
$
/HW T# W R2! R2EHDOLQHDUWUDQVIRUPDWLRQWKDWPDSV XD 3
4
$ LQWR
#4 1
$
DQGPDSV YD
#3 3
$ LQWR
#"1 3
$
8VHWKHIDFW WKDW T LVOLQHDUWRÀQGWKHLPDJHVXQGHU T RI 2X 3Y DQG 2XC 3Y
,QWURGXFWLRQWR/LQHDU7UDQVIRUPDWLRQV
7KHÀJXUHVKRZVYHFWRUV X Y DQG Z DORQJZLWKWKHLPDJHV T .X/ DQG T .Y/ XQGHUWKHDFWLRQRIDOLQHDUWUDQVIRUPDWLRQ T W R2! R2 &RS\WKLVÀJXUHFDUHIXOO\ DQGGUDZWKHLPDJH T .Z/ DVDFFXUDWHO\DVSRVVLEOH >+LQW )LUVW ZULWH Z DVD OLQHDUFRPELQDWLRQRI X DQG Y@
w u
v
T(v)
T(u)
x2 x2
x1 x1
/HW H1D
#1 0
$
H2D
#0 1
$
\1D
#2 5
$
DQG \2D
#"1 6
$
DQG OHW T W R2! R2EHDOLQHDUWUDQVIRUPDWLRQWKDWPDSV H1LQWR
\1DQGPDSV H2LQWR \2 )LQGWKHLPDJHVRI
# 5
"3
$ DQG
#x1
x2
$
/HW [D
#x1
x2
$
Y1D
#"3 5
$
DQG Y2D
# 7
"2
$
DQG OHW T W R2! R2 EHDOLQHDUWUDQVIRUPDWLRQWKDWPDSV [ LQWR x1Y1C x2Y2 )LQGDPDWUL[ A VXFKWKDW T .[/ LV A[ IRUHDFK [
,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\
HDFKDQVZHU
D $ OLQHDUWUDQVIRUPDWLRQLVDVSHFLDOW\SHRIIXQFWLRQ
E ,I A LVD 3# 5 PDWUL[DQG T LVDWUDQVIRUPDWLRQGHÀQHG E\ T .[/D A[ WKHQWKHGRPDLQRI T LV R3
F ,I A LVDQ m# n PDWUL[ WKHQWKHUDQJHRIWKHWUDQVIRUPD
WLRQ [7! A[ LV Rm
G (YHU\OLQHDUWUDQVIRUPDWLRQLVDPDWUL[WUDQVIRUPDWLRQ
H $ WUDQVIRUPDWLRQ T LVOLQHDULIDQGRQO\LI T .c1Y1C c2Y2/D c1T .Y1/C c2T .Y2/
IRUDOO Y1DQG Y2LQWKHGRPDLQRI T DQGIRUDOOVFDODUV c1
DQG c2
D 7KHUDQJHRIWKHWUDQVIRUPDWLRQ [7! A[ LVWKHVHWRIDOO OLQHDUFRPELQDWLRQVRIWKHFROXPQVRI A
E (YHU\PDWUL[WUDQVIRUPDWLRQLVDOLQHDUWUDQVIRUPDWLRQ
F ,I T W Rn! RmLVDOLQHDUWUDQVIRUPDWLRQDQGLI F LVLQ Rm WKHQDXQLTXHQHVVTXHVWLRQLV´,V F LQWKHUDQJHRI T"µ
G $ OLQHDUWUDQVIRUPDWLRQSUHVHUYHVWKHRSHUDWLRQVRIYHFWRU DGGLWLRQDQGVFDODUPXOWLSOLFDWLRQ
H $ OLQHDUWUDQVIRUPDWLRQ T W Rn! Rm DOZD\VPDSVWKH RULJLQRIRnWRWKHRULJLQRIRm:
'HÀQH f W R ! R E\ f .x/ D mx C b
D 6KRZWKDW f LVDOLQHDUWUDQVIRUPDWLRQZKHQ bD 0
E )LQGDSURSHUW\RIDOLQHDUWUDQVIRUPDWLRQWKDWLVYLRODWHG ZKHQ b¤ 0
F :K\LV f FDOOHGDOLQHDUIXQFWLRQ"
$Q DIÀQHWUDQVIRUPDWLRQ T W Rn! RmKDVWKHIRUP T .[/D A[C E ZLWK A DQ m # n PDWUL[DQG E LQ Rm 6KRZ WKDW T LV QRW DOLQHDUWUDQVIRUPDWLRQZKHQ E¤ $IÀQH WUDQVIRUPDWLRQVDUHLPSRUWDQWLQFRPSXWHUJUDSKLFV
*LYHQ Y¤ DQG S LQ Rn WKHOLQHWKURXJK S LQWKHGLUHFWLRQRI Y KDVWKHSDUDPHWULFHTXDWLRQ [D S C tY 6KRZWKDWDOLQHDU WUDQVIRUPDWLRQ T W Rn! RnPDSVWKLVOLQHRQWRDQRWKHUOLQH RURQWRDVLQJOHSRLQWD GHJHQHUDWHOLQH
D 6KRZWKDWWKHOLQHWKURXJKYHFWRUV S DQG T LQRnPD\EH ZULWWHQLQWKHSDUDPHWULFIRUP [D .1 " t/S C tT 5HIHU WRWKHÀJXUHZLWK([HUFLVHVDQGLQ6HFWLRQ
E 7KHOLQHVHJPHQWIURP S WR T LVWKHVHWRISRLQWVRIWKH IRUP .1" t/S C tT IRU 0 ' t ' 1 DVVKRZQLQWKHÀJXUH EHORZ 6KRZWKDWDOLQHDUWUDQVIRUPDWLRQ T PDSVWKLV OLQHVHJPHQWRQWRDOLQHVHJPHQWRURQWRDVLQJOHSRLQW
(t = 0) p (1 – t)p + tq
(t = 1) q T(p) x T(x)
T(q)
/HW X DQG Y EHOLQHDUO\LQGHSHQGHQWYHFWRUVLQR3 DQGOHW P EHWKHSODQHWKURXJK X Y DQG 7KHSDUDPHWULFHTXDWLRQ RI P LV [D sX C tY ZLWK s; t LQ R 6KRZWKDWDOLQHDU WUDQVIRUPDWLRQ T W R3! R3PDSV P RQWRDSODQHWKURXJK
RURQWRDOLQHWKURXJK RURQWRMXVWWKHRULJLQLQR3 :KDW PXVWEHWUXHDERXW T .X/ DQG T .Y/ LQRUGHUIRUWKHLPDJHRI WKHSODQH P WREHDSODQH"
/HW X DQG Y EHYHFWRUVLQRn ,WFDQEHVKRZQWKDWWKHVHW P RI DOOSRLQWVLQWKHSDUDOOHORJUDPGHWHUPLQHGE\ X DQG Y KDVWKH IRUP aXC bY IRU 0 ' a ' 1 0 ' b ' 1 /HW T W Rn! Rm EHDOLQHDUWUDQVIRUPDWLRQ ([SODLQZK\WKHLPDJHRIDSRLQW LQ P XQGHUWKHWUDQVIRUPDWLRQ T OLHVLQWKHSDUDOOHORJUDP GHWHUPLQHGE\ T .X/ DQG T .Y/
/HW TW R2! R2 EHWKHOLQHDUWUDQVIRUPDWLRQWKDWUHÁHFWV HDFKSRLQWWKURXJKWKH x2D[LV 0DNHWZRVNHWFKHVVLPLODU WR )LJ WKDW LOOXVWUDWH SURSHUWLHV L DQG LL RI D OLQHDU WUDQVIRUPDWLRQ
6XSSRVHYHFWRUV Y1; : : : ;YpVSDQRn DQGOHW T W Rn! Rn EH D OLQHDU WUDQVIRUPDWLRQ 6XSSRVH T .Yi/D IRU i D 1; : : : ; p 6KRZWKDW T LVWKH]HURWUDQVIRUPDWLRQ 7KDWLV
VKRZWKDWLI [ LVDQ\YHFWRULQRn WKHQ T .[/D
/HW T W Rn! Rm EH D OLQHDU WUDQVIRUPDWLRQ DQG OHW fY1;Y2;Y3g EHDOLQHDUO\GHSHQGHQWVHWLQ Rn ([SODLQZK\
WKHVHWfT .Y1/; T .Y2/; T .Y3/g LVOLQHDUO\GHSHQGHQW
,Q([HUFLVHV² FROXPQYHFWRUVDUHZULWWHQDVURZV VXFKDV [D .x1; x2/ DQG T .[/ LVZULWWHQDV T .x1; x2/
6KRZ WKDW WKH WUDQVIRUPDWLRQ T GHÀQHG E\ T .x1; x2/D .x1" 2jx2j; x1" 4x2/LVQRWOLQHDU
6KRZ WKDW WKH WUDQVIRUPDWLRQ T GHÀQHG E\ T .x1; x2/D .x1" 2x2; x1" 3; 2x1" 5x2/LVQRWOLQHDU
$)"15&3 /LQHDU(TXDWLRQVLQ/LQHDU$OJHEUD
/HW T W R3! R3 EHWKHWUDQVIRUPDWLRQWKDWUHÁHFWVHDFK YHFWRU [D .x1; x2; x3/ WKURXJK WKH SODQH x3D 0 RQWR T .[/D .x1; x2;"x3/ 6KRZWKDW T LVDOLQHDUWUDQVIRUPD
WLRQ >6HH([DPSOHIRULGHDV@
/HW T W R3! R3 EHWKHWUDQVIRUPDWLRQWKDWSURMHFWVHDFK YHFWRU [D .x1; x2; x3/ RQWRWKHSODQH x2D 0 VR T .[/ D .x1; 0; x3/ 6KRZWKDW T LVDOLQHDUWUDQVIRUPDWLRQ
/HW T W Rn! RmEHDOLQHDUWUDQVIRUPDWLRQ 6XSSRVHfX; Yg LVDOLQHDUO\LQGHSHQGHQWVHW EXWfT .X/; T .Y/g LVDOLQHDUO\
GHSHQGHQWVHW 6KRZWKDW T .[/D KDVDQRQWULYLDOVROXWLRQ
>+LQW 8VH WKH IDFW WKDW c1T .X/C c2T .Y/D IRUVRPH ZHLJKWV c1DQG c2 QRWERWK]HUR@
>0@ ,Q([HUFLVHVDQG WKHJLYHQPDWUL[GHWHUPLQHVDOLQHDU WUDQVIRUPDWLRQ T )LQGDOO [ VXFKWKDW T .[/D
2 66 4
2 3 5 "5
"7 7 0 0
"3 4 1 3
"9 3 "6 "4 3 77
5
2 66 4
3 4 "7 0 5 "8 7 4 6 "8 6 4 9 "7 "2 0 3 77 5
>0@ /HW ED 2 66 4
8 7 5
"3 3 77
5 DQGOHW A EHWKHPDWUL[LQ([HUFLVH
,V E LQWKHUDQJHRIWKHWUDQVIRUPDWLRQ [7! A[" ,IVR ÀQG DQ [ ZKRVHLPDJHXQGHUWKHWUDQVIRUPDWLRQLV E
>0@ /HW ED 2 66 4
4
"4
"4
"7 3 77
5 DQGOHW A EHWKHPDWUL[LQ([HUFLVH
,V E LQWKHUDQJHRIWKHWUDQVIRUPDWLRQ [7! A[" ,IVR ÀQG DQ [ ZKRVHLPDJHXQGHUWKHWUDQVIRUPDWLRQLV E
SG .BTUFSJOH -JOFBS 5SBOTGPSNBUJPOT m
40-65*0/4 50 13"$5*$& 130#-&.4
A PXVWKDYHÀYHFROXPQVIRU A[ WREHGHÀQHG A PXVWKDYHWZRURZVIRUWKH
Au
u x2
x1 Av
v
Ax x
7KHWUDQVIRUPDWLRQ [7!A[
FRGRPDLQRI T WREHR2
3ORWVRPHUDQGRPSRLQWVYHFWRUVRQJUDSKSDSHUWRVHHZKDWKDSSHQV $ SRLQWVXFK DV .4; 1/ PDSVLQWR .4;"1/ 7KHWUDQVIRUPDWLRQ [ 7!A[ UHÁHFWVSRLQWVWKURXJKWKH xD[LVRU x1D[LV
/HW [D tX IRUVRPH t VXFKWKDW 0 ' t ' 1 6LQFH T LVOLQHDU T .tX/ D t T .X/ ZKLFK LVDSRLQWRQWKHOLQHVHJPHQWEHWZHHQ DQG T .X/
5)& ."53*9 0' " -*/&"3 53"/4'03."5*0/
:KHQHYHUDOLQHDUWUDQVIRUPDWLRQ T DULVHVJHRPHWULFDOO\RULVGHVFULEHGLQZRUGV ZH XVXDOO\ZDQWD´IRUPXODµIRU T .[/ 7KHGLVFXVVLRQWKDWIROORZVVKRZVWKDWHYHU\OLQHDU WUDQVIRUPDWLRQIURP Rn WR Rm LVDFWXDOO\DPDWUL[WUDQVIRUPDWLRQ [7!A[ DQGWKDW LPSRUWDQWSURSHUWLHVRI T DUHLQWLPDWHO\UHODWHGWRIDPLOLDUSURSHUWLHVRI A 7KHNH\WR ÀQGLQJ A LVWRREVHUYHWKDW T LVFRPSOHWHO\GHWHUPLQHGE\ZKDWLWGRHVWRWKHFROXPQV RIWKH n# n LGHQWLW\PDWUL[ In
&9".1-& 7KHFROXPQVRI I2D
#1 0
0 1
$
DUH H1D
#1 0
$
DQG H2D
#0 1
$
6XS
SRVH T LVDOLQHDUWUDQVIRUPDWLRQIURPR2LQWRR3VXFKWKDW
T .H1/D 2 4 5
"7 2
3
5 DQG T .H2/D 2 4"3
8 0
3 5
:LWKQRDGGLWLRQDOLQIRUPDWLRQ ÀQGDIRUPXODIRUWKHLPDJHRIDQDUELWUDU\ [ LQR2
x1 x2
e2 = 01
e1 = 10
7KH0DWUL[RID/LQHDU7UDQVIRUPDWLRQ
40-65*0/ :ULWH
[D
#x1 x2
$ D x1
#1 0
$ C x2
#0 1
$
D x1H1C x2H2
6LQFH T LVD OLQHDU WUDQVIRUPDWLRQ
T .[/D x1T .H1/C x2T .H2/
D x1
2 4 5
"7 2
3 5C x2
2 4"3
8 0
3 5D
2
4 5x1" 3x2
"7x1C 8x2
2x1C 0 3 5
7KHVWHSIURPHTXDWLRQWRHTXDWLRQH[SODLQVZK\NQRZOHGJHRI T .H1/DQG T .H2/LVVXIÀFLHQWWRGHWHUPLQH T .[/ IRUDQ\ [ 0RUHRYHU VLQFHH[SUHVVHV T .[/ DV DOLQHDUFRPELQDWLRQRIYHFWRUV ZHFDQSXWWKHVHYHFWRUVLQWRWKHFROXPQVRIDPDWUL[
ADQGZULWHDV
T .[/D%T .H1/ T .H2/& #x1
x2
$ D A[
5 ) & 0 3 & . /HW T W Rn! RmEHDOLQHDUWUDQVIRUPDWLRQ 7KHQWKHUHH[LVWVDXQLTXHPDWUL[
AVXFKWKDW
T .[/D A[ IRUDOO [ LQ Rn
,QIDFW A LVWKH m# n PDWUL[ZKRVH j WKFROXPQLVWKHYHFWRU T .Hj/ ZKHUH Hj LV WKH j WKFROXPQRIWKHLGHQWLW\PDWUL[LQRn
AD%T .H1/ ! ! ! T .Hn/&
1300' :ULWH [D In[D Œ H1 ! ! ! Hn![D x1H1C ! ! ! C xnHn DQGXVHWKHOLQHDULW\
RI T WRFRPSXWH
T .[/D T .x1H1C ! ! ! C xnHn/D x1T .H1/C ! ! ! C xnT .Hn/
D%T .H1/ ! ! ! T .Hn/&
2 64
x1 ::: xn
3 75D A[
7KHXQLTXHQHVVRI A LVWUHDWHGLQ([HUFLVH
7KHPDWUL[ A LQLVFDOOHGWKH VWDQGDUGPDWUL[IRUWKHOLQHDUWUDQVIRUPD
WLRQ T
:HNQRZQRZWKDWHYHU\OLQHDUWUDQVIRUPDWLRQIURPRnWRRm FDQEHYLHZHGDV DPDWUL[WUDQVIRUPDWLRQ DQGYLFHYHUVD 7KHWHUP OLQHDUWUDQVIRUPDWLRQ IRFXVHVRQD SURSHUW\RIDPDSSLQJ ZKLOH PDWUL[WUDQVIRUPDWLRQ GHVFULEHVKRZVXFKDPDSSLQJLV LPSOHPHQWHG DV([DPSOHV DQGLOOXVWUDWH
&9".1-& )LQGWKHVWDQGDUGPDWUL[ A IRUWKHGLODWLRQWUDQVIRUPDWLRQ T .[/D 3[
IRU [ LQR2
$)"15&3 /LQHDU(TXDWLRQVLQ/LQHDU$OJHEUD
40-65*0/ :ULWH
T .H1/D 3H1D
#3 0
$
DQG T .H2/D 3H2D
#0 3
$
# # AD
#3 0
0 3
$
&9".1-& /HW T W R2 ! R2 EHWKHWUDQVIRUPDWLRQWKDWURWDWHVHDFKSRLQWLQ R2 DERXWWKHRULJLQWKURXJKDQDQJOH ' ZLWKFRXQWHUFORFNZLVHURWDWLRQIRUDSRVLWLYH DQJOH :HFRXOGVKRZJHRPHWULFDOO\WKDWVXFKDWUDQVIRUPDWLRQLVOLQHDU 6HH)LJ LQ 6HFWLRQ )LQGWKHVWDQGDUGPDWUL[ A RIWKLVWUDQVIRUPDWLRQ
40-65*0/
#1 0
$
URWDWHVLQWR
#FRV ' VLQ '
$
DQG
#0 1
$
URWDWHVLQWR
#" VLQ ' FRV '
$
6HH)LJ
%\7KHRUHP
AD
#FRV ' " VLQ ' VLQ ' FRV '
$
([DPSOHLQ6HFWLRQ LVDVSHFLDOFDVHRIWKLVWUDQVIRUPDWLRQ ZLWK 'D "=2
(– sin ϕ, cos ϕ)
(cos ϕ, sin ϕ) (1, 0) (0, 1)
ϕ
ϕ x1
x2
'*(63& $ URWDWLRQWUDQVIRUPDWLRQ
*HRPHWULF/LQHDU7UDQVIRUPDWLRQVRI R
2([DPSOHVDQGLOOXVWUDWHOLQHDUWUDQVIRUPDWLRQVWKDWDUHGHVFULEHGJHRPHWULFDOO\
7DEOHV ² LOOXVWUDWH RWKHU FRPPRQ JHRPHWULF OLQHDU WUDQVIRUPDWLRQV RI WKH SODQH
%HFDXVHWKHWUDQVIRUPDWLRQVDUHOLQHDU WKH\DUHGHWHUPLQHGFRPSOHWHO\E\ZKDWWKH\
GRWRWKHFROXPQVRI I2 ,QVWHDGRIVKRZLQJRQO\WKHLPDJHVRI H1 DQG H2 WKHWDEOHV VKRZZKDWDWUDQVIRUPDWLRQGRHVWRWKHXQLWVTXDUH)LJ
2WKHU WUDQVIRUPDWLRQV FDQ EH FRQVWUXFWHG IURP WKRVH OLVWHG LQ 7DEOHV ² E\
DSSO\LQJRQHWUDQVIRUPDWLRQDIWHUDQRWKHU )RULQVWDQFH DKRUL]RQWDOVKHDUFRXOGEH IROORZHGE\DUHÁHFWLRQLQWKH x2D[LV 6HFWLRQ ZLOOVKRZWKDWVXFKD FRPSRVLWLRQ RIOLQHDUWUDQVIRUPDWLRQVLVOLQHDU $OVR VHH([HUFLVH
x1 x2
01
10
'*(63& 7KHXQLWVTXDUH
([LVWHQFHDQG8QLTXHQHVV4XHVWLRQV
7KHFRQFHSWRIDOLQHDUWUDQVIRUPDWLRQSURYLGHVDQHZZD\WRXQGHUVWDQGWKHH[LVWHQFH DQGXQLTXHQHVVTXHVWLRQVDVNHGHDUOLHU 7KHWZRGHÀQLWLRQVIROORZLQJ7DEOHV²JLYH WKHDSSURSULDWHWHUPLQRORJ\IRUWUDQVIRUPDWLRQV
7KH0DWUL[RID/LQHDU7UDQVIRUPDWLRQ
5"#-& 3FáFDUJPOT
7UDQVIRUPDWLRQ ,PDJHRIWKH8QLW6TXDUH 6WDQGDUG0DWUL[
5HÁHFWLRQWKURXJK WKH x1D[LV
x1 x2
–10
10
#1 0
0 "1
$
5HÁHFWLRQWKURXJK WKH x2D[LV
x1 x2
–10
01
#"1 0
0 1
$
5HÁHFWLRQWKURXJK WKHOLQH x2D x1
x1 x2
x2 = x1
10
01
#0 1
1 0
$
5HÁHFWLRQWKURXJK WKHOLQH x2D "x1
x1 x2 = –x1 x2
–10
–10
# 0 "1
"1 0
$
5HÁHFWLRQWKURXJK WKHRULJLQ
x1 x2
–10
–10
#"1 0 0 "1
$
$)"15&3 /LQHDU(TXDWLRQVLQ/LQHDU$OJHEUD
5"#-& $POUSBDUJPOT BOE &YQBOTJPOT
7UDQVIRUPDWLRQ ,PDJHRIWKH8QLW6TXDUH 6WDQGDUG0DWUL[
+RUL]RQWDO FRQWUDFWLRQ DQGH[SDQVLRQ
x1 x2
01
0k k > 1 x1
x2
01
k0 0 < k < 1
#k 0
0 1
$
9HUWLFDO FRQWUDFWLRQ DQGH[SDQVLRQ
10
0k
x1 x2
10
0k
x1 x2
k > 1 0 < k < 1
#1 0
0 k
$
5"#-& 4IFBST
7UDQVIRUPDWLRQ ,PDJHRIWKH8QLW6TXDUH 6WDQGDUG0DWUL[
+RUL]RQWDOVKHDU
x1 x2
k < 0
k x1
x2
k > 0 k k
1
k 1
1 0
1 0
#1 k
0 1
$
9HUWLFDOVKHDU
01
01
1k
1k
x1 x2
x1 x2
k < 0 k > 0
k
k
#1 0
k 1
$
7KH0DWUL[RID/LQHDU7UDQVIRUPDWLRQ
5"#-& 1SPKFDUJPOT
7UDQVIRUPDWLRQ ,PDJHRIWKH8QLW6TXDUH 6WDQGDUG0DWUL[
3URMHFWLRQRQWR WKH x1D[LV
00
10
x1
x2 #
1 0
0 0
$
3URMHFWLRQRQWR WKH x2D[LV
00
01
x1
x2 #
0 0
0 1
$
% & ' * / * 5 * 0 / $ PDSSLQJ T W Rn! Rm LVVDLGWREH RQWRRmLIHDFK E LQRm LVWKHLPDJHRI DWOHDVWRQH [ LQRn
(TXLYDOHQWO\ T LVRQWRRmZKHQWKHUDQJHRI T LVDOORIWKHFRGRPDLQRm 7KDWLV
T PDSVRnRQWRRmLI IRUHDFK E LQWKHFRGRPDLQRm WKHUHH[LVWVDWOHDVWRQHVROXWLRQ RI T .[/D E ´'RHV T PDS RnRQWRRm"µ LVDQH[LVWHQFHTXHVWLRQ 7KHPDSSLQJ T LV QRW RQWRZKHQWKHUHLVVRPH E LQRmIRUZKLFKWKHHTXDWLRQ T .[/D E KDVQRVROXWLRQ
6HH)LJ
!m
Domain
Range
T T
!n !n
Domain
!m Range
T is not onto !m T is onto !m
'*(63& ,VWKHUDQJHRI T DOORIRm"
% & ' * / * 5 * 0 / $ PDSSLQJ T W Rn! RmLVVDLGWREH RQHWRRQH LIHDFK E LQRmLVWKHLPDJH RI DWPRVWRQH [ LQRn
$)"15&3 /LQHDU(TXDWLRQVLQ/LQHDU$OJHEUD
(TXLYDOHQWO\ T LVRQHWRRQHLI IRUHDFK E LQRm WKHHTXDWLRQ T .[/D E KDVHLWKHU DXQLTXHVROXWLRQRUQRQHDWDOO ´,V T RQHWRRQH"µ LVDXQLTXHQHVVTXHVWLRQ 7KH PDSSLQJ T LV QRW RQHWRRQHZKHQVRPH E LQRmLVWKHLPDJHRIPRUHWKDQRQHYHFWRU LQRn ,IWKHUHLVQRVXFK E WKHQ T LVRQHWRRQH 6HH)LJ
Domain
Range
Domain
Range
0
T
T is not one-to-one
0 0
T is one-to-one T
0
!n !m !n !m
'*(63& ,VHYHU\ E WKHLPDJHRIDWPRVWRQHYHFWRU"
7KH SURMHFWLRQWUDQVIRUPDWLRQVVKRZQLQ7DEOHDUH QRW RQHWRRQHDQGGR QRW PDS
SG .BTUFSJOH &YJTUFODF
BOE 6OJRVFOFTT m R2 RQWRR2 7KHWUDQVIRUPDWLRQVLQ7DEOHV DQGDUHRQHWRRQH DQG GRPDSR2 RQWRR2 2WKHUSRVVLELOLWLHVDUHVKRZQLQWKHWZRH[DPSOHVEHORZ
([DPSOHDQGWKHWKHRUHPVWKDWIROORZVKRZKRZWKHIXQFWLRQSURSHUWLHVRIEHLQJ RQHWRRQHDQGPDSSLQJRQWRDUHUHODWHGWRLPSRUWDQWFRQFHSWVVWXGLHGHDUOLHULQWKLV FKDSWHU
&9".1-& /HW T EHWKHOLQHDUWUDQVIRUPDWLRQZKRVHVWDQGDUGPDWUL[LV
AD 2
41 "4 8 1 0 2 "1 3
0 0 0 5
3 5
'RHV T PDSR4RQWRR3" ,V T DRQHWRRQHPDSSLQJ"
40-65*0/ 6LQFH A KDSSHQVWREHLQHFKHORQIRUP ZHFDQVHHDWRQFHWKDW A KDVD SLYRWSRVLWLRQLQHDFKURZ %\7KHRUHPLQ6HFWLRQ IRUHDFK E LQR3 WKHHTXDWLRQ A[D E LVFRQVLVWHQW ,QRWKHUZRUGV WKHOLQHDUWUDQVIRUPDWLRQ T PDSV R4LWVGRPDLQ RQWRR3 +RZHYHU VLQFHWKHHTXDWLRQ A[D E KDVDIUHHYDULDEOHEHFDXVHWKHUHDUH IRXUYDULDEOHVDQGRQO\WKUHHEDVLFYDULDEOHV HDFK E LVWKHLPDJHRIPRUHWKDQRQH [
7KDWLV T LV QRW RQHWRRQH
5 ) & 0 3 & . /HW T W Rn! RmEHDOLQHDUWUDQVIRUPDWLRQ 7KHQ T LVRQHWRRQHLIDQGRQO\LI WKHHTXDWLRQ T .[/D KDVRQO\WKHWULYLDOVROXWLRQ
1300' 6LQFH T LVOLQHDU T ./D ,I T LVRQHWRRQH WKHQWKHHTXDWLRQ T .[/ D KDVDWPRVWRQHVROXWLRQDQGKHQFHRQO\WKHWULYLDOVROXWLRQ ,I T LVQRWRQHWRRQH WKHQ WKHUHLVD E WKDWLVWKHLPDJHRIDWOHDVWWZRGLIIHUHQWYHFWRUVLQRn³VD\ X DQG Y 7KDW LV T .X/D E DQG T .Y/ D E %XWWKHQ VLQFH T LVOLQHDU
T .X" Y/ D T .X/ " T .Y/ D E " E D
7KHYHFWRU X" Y LVQRW]HUR VLQFH X ¤ Y +HQFHWKHHTXDWLRQ T .[/ D KDVPRUHWKDQ RQHVROXWLRQ 6R HLWKHUWKHWZRFRQGLWLRQVLQWKHWKHRUHPDUHERWKWUXHRUWKH\DUHERWK IDOVH
7KH0DWUL[RID/LQHDU7UDQVIRUPDWLRQ
5 ) & 0 3 & . /HW T W Rn! RmEHDOLQHDUWUDQVIRUPDWLRQDQGOHW A EHWKHVWDQGDUGPDWUL[IRU T 7KHQ
D T PDSVRnRQWRRmLIDQGRQO\LIWKHFROXPQVRI A VSDQRm
E T LVRQHWRRQHLIDQGRQO\LIWKHFROXPQVRI A DUHOLQHDUO\LQGHSHQGHQW
1300'
D %\7KHRUHPLQ6HFWLRQ WKHFROXPQVRI A VSDQRm LIDQGRQO\LIIRUHDFK E LQRmWKHHTXDWLRQ A[D E LVFRQVLVWHQW³LQRWKHUZRUGV LIDQGRQO\LIIRUHYHU\ E
WKHHTXDWLRQ T .[/D E KDVDWOHDVWRQHVROXWLRQ 7KLVLVWUXHLIDQGRQO\LI T PDSV RnRQWRRm
E 7KHHTXDWLRQV T .[/D DQG A[ D DUHWKHVDPHH[FHSWIRUQRWDWLRQ 6R E\
7KHRUHP T LVRQHWRRQHLIDQGRQO\LI A[D KDVRQO\WKHWULYLDOVROXWLRQ
7KLVKDSSHQVLIDQGRQO\LIWKHFROXPQVRI A DUHOLQHDUO\LQGHSHQGHQW DVZDVDOUHDG\
QRWHGLQWKHER[HGVWDWHPHQWLQ6HFWLRQ
6WDWHPHQWDLQ7KHRUHPLVHTXLYDOHQWWRWKHVWDWHPHQW´T PDSVRnRQWRRm LIDQGRQO\LIHYHU\YHFWRULQRm LVDOLQHDUFRPELQDWLRQRIWKHFROXPQVRI Aµ 6HH 7KHRUHPLQ6HFWLRQ
,QWKHQH[WH[DPSOHDQGLQVRPHH[HUFLVHVWKDWIROORZ FROXPQYHFWRUVDUHZULWWHQLQ URZV VXFKDV [D .x1; x2/ DQG T .[/ LVZULWWHQDV T .x1; x2/LQVWHDGRIWKHPRUHIRUPDO T ..x1; x2//
&9".1-& /HW T .x1; x2/D .3x1C x2 5x1C 7x2 x1C 3x2/ 6KRZWKDW T LVD RQHWRRQHOLQHDUWUDQVIRUPDWLRQ 'RHV T PDSR2RQWRR3"
40-65*0/ :KHQ [ DQG T .[/ DUHZULWWHQDVFROXPQYHFWRUV \RXFDQGHWHUPLQHWKH VWDQGDUGPDWUL[RI T E\LQVSHFWLRQ YLVXDOL]LQJWKHURZ²YHFWRUFRPSXWDWLRQRIHDFK HQWU\LQ A[
T .[/D 2
43x1C x2 5x1C 7x2 x1C 3x2
3 5D
2 4" "
" "
" "
3 5#
x1
x2
$ D
2 43 1
5 7
1 3
3 5#
x1
x2
$
A
6R T LVLQGHHGDOLQHDUWUDQVIRUPDWLRQ ZLWKLWVVWDQGDUGPDWUL[ A VKRZQLQ 7KH FROXPQVRI A DUHOLQHDUO\LQGHSHQGHQWEHFDXVHWKH\DUHQRWPXOWLSOHV %\7KHRUHP
E T LVRQHWRRQH 7RGHFLGHLI T LVRQWRR3 H[DPLQHWKHVSDQRIWKHFROXPQVRI A 6LQFH A LV 3# 2 WKHFROXPQVRI A VSDQ R3LIDQGRQO\LI A KDVSLYRWSRVLWLRQV
E\7KHRUHP 7KLVLVLPSRVVLEOH VLQFH A KDVRQO\FROXPQV 6RWKHFROXPQVRI A GR QRWVSDQR3 DQGWKHDVVRFLDWHGOLQHDUWUDQVIRUPDWLRQLVQRWRQWRR3
T T
e2 e1
x3
x1 x1 x2
a2 a1 Span{a1, a2}
7KHWUDQVIRUPDWLRQ T LVQRW RQWRR3
13"$5*$& 130#-&.
/HW T W R2! R2EHWKHWUDQVIRUPDWLRQWKDWÀUVWSHUIRUPVDKRUL]RQWDOVKHDUWKDWPDSV H2LQWR H2" :5H1EXWOHDYHV H1 XQFKDQJHGDQGWKHQUHÁHFWVWKHUHVXOWWKURXJKWKH x2 D[LV $VVXPLQJWKDW T LVOLQHDU ÀQGLWVVWDQGDUGPDWUL[ >+LQW 'HWHUPLQHWKHÀQDO ORFDWLRQRIWKHLPDJHVRI H1DQG H2@
$)"15&3 /LQHDU(TXDWLRQVLQ/LQHDU$OJHEUD
&9&3$*4&4
,Q([HUFLVHV² DVVXPHWKDW T LVDOLQHDUWUDQVIRUPDWLRQ )LQG WKHVWDQGDUGPDWUL[RI T
T W R2! R4 T .H1/D .3; 1; 3; 1/ DQG T .H2/D ."5; 2; 0; 0/
ZKHUH H1D .1; 0/ DQG H2D .0; 1/
T W R3! R2 T .H1/D .1; 4/ T .H2/D ."2; 9/ DQG T .H3/D .3; "8/ ZKHUH H1 H2 DQG H3 DUHWKHFROXPQVRI WKH 3# 3 LGHQWLW\PDWUL[
T W R2! R2LVDYHUWLFDOVKHDUWUDQVIRUPDWLRQWKDWPDSV H1
LQWR H1" 3H2 EXWOHDYHV H2XQFKDQJHG
T W R2! R2LVDKRUL]RQWDOVKHDUWUDQVIRUPDWLRQWKDWOHDYHV H1XQFKDQJHGDQGPDSV H2LQWR H2C 2H1
T W R2! R2URWDWHVSRLQWVDERXWWKHRULJLQWKURXJK "=2 UDGLDQVFRXQWHUFORFNZLVH
T W R2! R2 URWDWHV SRLQWV DERXW WKH RULJLQ WKURXJK
"3"=2 UDGLDQVFORFNZLVH
T W R2! R2 ÀUVW URWDWHV SRLQWV WKURXJK "3"=4 UDGLDQV FORFNZLVHDQGWKHQUHÁHFWVSRLQWVWKURXJKWKHKRUL]RQWDO x1D[LV >+LQW T .H1/D ."1=p
2; 1=p 2/@
T W R2! R2 ÀUVWSHUIRUPVDKRUL]RQWDOVKHDUWKDWWUDQV
IRUPV H2 LQWR H2C 2H1 OHDYLQJ H1XQFKDQJHGDQGWKHQUH
ÁHFWVSRLQWVWKURXJKWKHOLQH x2D "x1
T W R2! R2ÀUVWUHÁHFWVSRLQWVWKURXJKWKHKRUL]RQWDO x1 D[LVDQGWKHQURWDWHVSRLQWV""=2 UDGLDQV
T W R2! R2ÀUVWUHÁHFWVSRLQWVWKURXJKWKHKRUL]RQWDO x1 D[LVDQGWKHQUHÁHFWVSRLQWVWKURXJKWKHOLQH x2D x1
$ OLQHDU WUDQVIRUPDWLRQ T W R2! R2 ÀUVW UHÁHFWV SRLQWV WKURXJKWKH x1D[LVDQGWKHQUHÁHFWVSRLQWVWKURXJKWKH x2 D[LV 6KRZWKDW T FDQDOVREHGHVFULEHGDVDOLQHDUWUDQVIRU
PDWLRQWKDWURWDWHVSRLQWVDERXWWKHRULJLQ :KDWLVWKHDQJOH RIWKDWURWDWLRQ"
6KRZ WKDW WKH WUDQVIRUPDWLRQ LQ ([HUFLVH LV PHUHO\ D URWDWLRQDERXWWKHRULJLQ :KDWLVWKHDQJOHRIWKHURWDWLRQ"
/HW T W R2! R2EHWKHOLQHDUWUDQVIRUPDWLRQVXFKWKDW T .H1/ DQG T .H2/DUHWKHYHFWRUVVKRZQLQWKHÀJXUH 8VLQJWKH ÀJXUH VNHWFKWKHYHFWRU T .2; 1/
T(e1) T(e2)
x1 x2
/HW T W R2! R2 EHDOLQHDUWUDQVIRUPDWLRQZLWKVWDQGDUG PDWUL[ AD Œ D1 D2! ZKHUH D1 DQG D2 DUHVKRZQLQWKH ÀJXUHDWWKHWRSRIFROXPQ 8VLQJWKHÀJXUH GUDZWKH LPDJHRI
# 1
"2
$
XQGHUWKHWUDQVIRUPDWLRQ T
x2
x1 a1
a2
,Q([HUFLVHVDQG ÀOOLQWKHPLVVLQJHQWULHVRIWKHPDWUL[
DVVXPLQJWKDWWKHHTXDWLRQKROGVIRUDOOYDOXHVRIWKHYDULDEOHV
2
4‹ ‹ ‹
‹ ‹ ‹
‹ ‹ ‹
3 5 2 4x1
x2
x3
3 5 D
2
4 2x1" 4x2 x1" x3
"x2C 3x3
3 5
2
4‹ ‹
‹ ‹
‹ ‹
3 5#
x1
x2
$ D
2
43x1" 2x2 x1C 4x2
x2
3 5
,Q([HUFLVHV² VKRZWKDW T LVDOLQHDUWUDQVIRUPDWLRQE\
ÀQGLQJDPDWUL[WKDWLPSOHPHQWVWKHPDSSLQJ 1RWHWKDW x1; x2; : : : DUHQRWYHFWRUVEXWDUHHQWULHVLQYHFWRUV
T .x1; x2; x3; x4/D .x1C 2x2; 0; 2x2C x4; x2" x4/
T .x1; x2/D .x1C 4x2; 0; x1" 3x2; x1/
T .x1; x2; x3/D .x1" 5x2C 4x3; x2" 6x3/
T .x1; x2; x3; x4/D 3x1C 4x3" 2x41RWLFH TW R4! R/
/HW T W R2! R2 EH D OLQHDU WUDQVIRUPDWLRQ VXFK WKDW T .x1; x2/D .x1C x2; 4x1C 5x2/ )LQG [ VXFKWKDW T .[/D .3; 8/
/HW T W R2! R3 EH D OLQHDU WUDQVIRUPDWLRQ ZLWK T .x1; x2/D .2x1" x2;"3x1C x2; 2x1" 3x2/ )LQG [ VXFK WKDW T .[/D .0; "1; "4/
,Q([HUFLVHVDQG PDUNHDFKVWDWHPHQW7UXHRU)DOVH -XVWLI\
HDFKDQVZHU
D $ OLQHDUWUDQVIRUPDWLRQ T W Rn! RmLVFRPSOHWHO\GH
WHUPLQHGE\LWVHIIHFWRQWKHFROXPQVRIWKH n# n LGHQWLW\
PDWUL[
E ,I T W R2! R2 URWDWHVYHFWRUVDERXWWKHRULJLQWKURXJK DQDQJOH ' WKHQ T LVDOLQHDUWUDQVIRUPDWLRQ
F :KHQWZROLQHDUWUDQVIRUPDWLRQVDUHSHUIRUPHGRQHDIWHU DQRWKHU WKHFRPELQHGHIIHFWPD\QRWDOZD\VEHDOLQHDU WUDQVIRUPDWLRQ
G $ PDSSLQJ T W Rn! RmLVRQWRRmLIHYHU\YHFWRU [ LQ RnPDSVRQWRVRPHYHFWRULQRm
H ,I A LVD 3# 2 PDWUL[ WKHQWKHWUDQVIRUPDWLRQ [ 7! A[
FDQQRWEHRQHWRRQH
D ,I A LVD 4# 3 PDWUL[ WKHQWKHWUDQVIRUPDWLRQ [ 7! A[
PDSVR3RQWRR4