• 沒有找到結果。

/LQHDU7UDQVIRUPDWLRQV

在文檔中 /LQHDU(TXDWLRQVLQ /LQHDU$OJHEUD (頁 65-84)

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 / $ WUDQVIRUPDWLRQ RUPDSSLQJ 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\ L VD\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\  IROORZVIURPFRQGLWLRQ LL LQWKHGHÀQLWLRQ EHFDXVH T ./D T .0X/ D 0T .X/D  3URSHUW\  UHTXLUHVERWK L DQG LL 

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 5HSHDWHGDSSOLFDWLRQRI  SURGXFHVDXVHIXOJHQHUDOL]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 WUDQVIRUPVDOLVWRISURGXFWLRQTXDQWLWLHV PHDVXUHGLQGROODUV LQWRD 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

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 RURQWRDVLQJOHSRLQW D 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

 3ORWVRPHUDQGRPSRLQWV YHFWRUV RQJUDSKSDSHUWRVHHZKDWKDSSHQV $ SRLQWVXFK DV .4; 1/ PDSVLQWR .4;"1/ 7KHWUDQVIRUPDWLRQ [ 7!A[ UHÁHFWVSRLQWVWKURXJKWKH xD[LV RU 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

7KHVWHSIURPHTXDWLRQ  WRHTXDWLRQ  H[SODLQVZK\NQRZOHGJHRI T .H1/DQG T .H2/LVVXIÀFLHQWWRGHWHUPLQH T .[/ IRUDQ\ [ 0RUHRYHU VLQFH  H[SUHVVHV T .[/ DV DOLQHDUFRPELQDWLRQRIYHFWRUV ZHFDQSXWWKHVHYHFWRUVLQWRWKHFROXPQVRIDPDWUL[

ADQGZULWH  DV

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 ΠH1 ! ! ! 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 LQ  LVFDOOHGWKH 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 R4 LWVGRPDLQ RQWRR3 +RZHYHU VLQFHWKHHTXDWLRQ A[D E KDVDIUHHYDULDEOH EHFDXVHWKHUHDUH 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[HGVWDWHPHQW  LQ6HFWLRQ 

6WDWHPHQW D LQ7KHRUHPLVHTXLYDOHQWWRWKHVWDWHPHQW´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" :5H1 EXWOHDYHV H1 XQFKDQJHG DQGWKHQUHÁHFWVWKHUHVXOWWKURXJKWKH 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! R2URWDWHVSRLQWV DERXWWKHRULJLQ WKURXJK "=2 UDGLDQV FRXQWHUFORFNZLVH 

 T W R2! R2 URWDWHV  SRLQWV  DERXW  WKH  RULJLQ  WKURXJK

"3"=2 UDGLDQV FORFNZLVH 

 T W R2! R2 ÀUVW  URWDWHV  SRLQWV  WKURXJK "3"=4 UDGLDQV FORFNZLVH DQGWKHQUHÁHFWVSRLQWVWKURXJKWKHKRUL]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 H1XQFKDQJHG DQGWKHQUH

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" 2x4 1RWLFH 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

在文檔中 /LQHDU(TXDWLRQVLQ /LQHDU$OJHEUD (頁 65-84)

相關文件