Lineer Cebir - Linear Equations and Matrices

8/10/2019 Lineer Cebir - Linear Equations and Matrices

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F L N Z X B _ 6

Imkbnr Bquntmeks nkh

Dntrmfbs

Mk tlms flnptbr wb mktrehufb dntrmfbs vmn tlb tlbery ec smduitnkbeus imkbnrbquntmeks. Xlms dbtleh lns tlb nhvnktn`b ec ibnhmk` mk n knturni wny te tlbfekfbpt ec tlb rbhufbh rew-bflbiekcerd ec n dntrmx. Mk nhhmtmek, wb wmii cer-duintbsedb ec tlb onsmf rbsuits hbnimk` wmtl tlb bxmstbkfb nkh ukmqubkbss ec

systbds ec imkbnr bquntmeks. Mk Flnptbr 2 wb wmii nrrmvb nt tlb sndb dntrmxni`borncred tlb vmbwpemkt ec imkbnr trnkscerdntmeks.

6.> V]VXBDV EC IMKBN_ B^RNXMEKV

Ibt n, . . . , n, y ob bibdbkts ec n cmbih C, nkh ibt x, . . . , x ob ukakewks(nise fniibh vnrmnoibser mkhbtbrdmkntbs). Xlbk nk bquntmek ec tlb cerd

n x + ~ ~ ~ + n x 4 y

ms fniibh n imkbnr bquntmek mk k ukakewks(evbrC). Xlb sfninrs n nrb fniibhtlb febccmfmbkts ec tlb ukakewks, nkh y ms fniibh tlb fekstnkt tbrd ec tlb

bquntmek. N vbfter (f, . . . , f) !Ckms fniibh n seiutmek vbfterec tlms bqun-tmekmc nkh ekiy mc

n> f>+ ~ ~ ~ + nkfk 4 y


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IMKBN_ B^RNXMEKV NKH DNX_MFBV>>3

mk wlmfl fnsb wb sny tlnt (f, . . . , f) sntmscmbstlb bquntmek. Xlb sbt ec niisufl seiutmeks ms fniibh tlb seiutmek sbt(er tlb `bkbrni seiutmek).

Kew feksmhbr tlb ceiiewmk` systbd ec d imkbnr bquntmeks mk kukakewks7

n>>x>+!+ n

>kxk 4 y

>

n?>x>+!+ n?kxk 4 y?

" " "

nd>x>+!+ n

dkxk 4 y

d

Pb noorbvmntb tlms systbd oy

nmgxg 4ym , m 4>,,d .

g4>

k

!

Mc wb ibt Vmhbketb tlb seiutmek sbt ec tlb bquntmek nx 4 y cer bnfl m, tlbktlb seiutmek sbt V ec tlb systbd ms `mvbk oy tlb mktbrsbftmek V 4 V. Mk etlbr

werhs, mc (f, . . . , f) !Ckms n seiutmek ec tlb systbd ec bquntmeks, tlbk mt ms nseiutmek ec bnfl ec tlb d bquntmeks mk tlb systbd.

Bxndpib 6.> Feksmhbr tlms systbd ec twe bquntmeks mk tlrbb ukakewks evbrtlb rbni cmbih 7

?x>! 6x

?+ x

643

x>+ 2x

?! ?x

64>?

Xlb vbfter (6, >, 6) !6ms ket n seiutmek ec tlms systbd obfnusb

?(6) - 6(>) + 6 4 3wlmib

6 + 2(>) - ?(6) 4 ? " >? .

Lewbvbr, tlb vbfter (2, >, ->) !6ms n seiutmek smkfb

?(2) - 6(>) + (->) 4 3nkh

2 + 2(>) - ?(->) 4 >? . #

Nssefmntbh wmtl n systbd ec imkbnr bquntmeks nrb twe rbftnk`uinr nrrnys ecbibdbkts ec C tlnt turk eut te ob ec `rbnt tlberbtmfni ns wbii ns prnftmfnism`kmcmfnkfb. Cer tlb systbd nx 4 y, wb hbcmkb tlb dntrmx ec febccmfmbktsN ns tlb nrrny


3/55

6.> V]VXBDV EC IMKBN_ B^RNXMEKV >>0

N 4

n>>

n>?

! n>k

n?>

n??

! n?k

" " "

nd> nd? ! ndk

!

"

####

$

%

&&&&

nkh tlb nu`dbktbh dntrmxns tlb nrrny nu` N `mvbk oy

nu`

N 4

n>>

n>?

! n>k y>

n?>

n??

! n?k y?

" " " "

nd> nd? ! ndk yk

!

"

####

$

%

&&&&

.

Mk `bkbrni, wb wmii usb tlb tbrd dntrmx te hbketb nky nrrny sufl ns tlbnrrny N slewk noevb. Xlms dntrmx lns d rewsnkh k feiudks, nkh lbkfb msrbcbrrbh te ns nk d x k dntrmx, er n dntrmx ec smzbd x k. Oy fekvbktmek, nkbibdbkt n !Cec N ms inobibh wmtl tlb cmrst mkhbx rbcbrrmk` te tlb rew nkh tlbsbfekh mkhbx rbcbrrmk` te tlb feiudk. Xlb sfninr n ms usuniiy fniibh tlb m, gtlbktry (er bibdbkt) ec tlb dntrmx N. Pb wmii crbqubktiy hbketb tlb dntrmx Noy tlb sydoei (n).

Nketlbr rntlbr `bkbrni wny te hbcmkb n dntrmx ms ns n dnppmk` cred n suo-sbtec nii erhbrbh pnmrs ec pesmtmvb mktb`brs mkte tlb cmbih C. Mk etlbr werhs, wbhbcmkbtlb dnppmk` N oy N(m, g) 4 n cer bvbry > m d nkh > g k. Mk tlms

sbksb, n dntrmx ms nftuniiy n dnppmk`, nkh tlb d x k nrrny wrmttbk noevb ms gustn rbprbsbktntmek ec tlms dnppmk`.Obcerb prefbbhmk` wmtl tlb `bkbrni tlbery, ibt us `mvb n spbfmcmf bxndpib

hbdekstrntmk` lew te seivb n systbd ec imkbnr bquntmeks.

Bxndpib 6.? Ibt us nttbdpt te seivb tlb ceiiewmk` systbd ec imkbnr bqun-tmeks7

?x>+ x

?! ?x

6 4 !6

x>! 6x

?+ x

6 4 ! x

?! ?x

64 6

Xlnt eur npprenfl ms vnimh mk `bkbrni wmii ob prevbh mk eur cmrst tlberbdobiew.

Duitmpiy tlb cmrst bquntmek oy >/? te `bt tlb febccmfmbkt ec x bquni te >7


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IMKBN_ B^RNXMEKV NKH DNX_MFBV>> + (> / ?)x?! x6 4 !6 / ?

x>! 6x? + x6 4 ! x?! ?x6 4 6

Duitmpiy tlb cmrst bquntmek oy -> nkh nhh mt te tlb sbfekh te eotnmk n kbw sbf-ekhbquntmek, tlbk duitmpiy tlb cmrst oy -8 nkh nhh mt te tlb tlmrh te eotnmk nkbw tlmrh bquntmek7

x> + (> / ?)x?! x6 4 !6 / ?

!(0 / ?)x? + ?x6 4>= / ?

!6x?! ?x6 4 =

Duitmpiy tlb sbfekh oy -?/0 te `bt tlb febccmfmbkt ec x bquni te >, tlbk dui-tmpiytlms kbw sbfekh bquntmek oy 6 nkh nhh te tlb tlmrh7

x> +

(> / ?)x?!

x6 4 !

6 / ? x?! (8 / 0)x6 4!>= / 0

(? / 0)x6 4 3 / 0

Duitmpiy tlb tlmrh oy 0/?, tlbk nhh 8/0 tmdbs tlms kbw bquntmek te tlb sbfekh7

x> + (> / ?)x?! x6 4!6 / ?

x? 4 !>

x6 4 6

Nhh tlb tlmrh bquntmek te tlb cmrst, tlbk nhh ->/? tmdbs tlb sbfekh bquntmek tetlb kbw cmrst te eotnmk

x> 4

?

x? 4 !>

x6 4 6

Xlms ms kew n seiutmek ec eur systbd ec bquntmeks. Plmib tlms systbd feuihlnvb obbk seivbh mk n derb hmrbft dnkkbr, wb wnktbh te miiustrntb tlb systbd-ntmfnpprenfl tlnt wmii ob kbbhbh obiew. #

Xwe systbds ec imkbnr bquntmeks nrb snmh te ob bqumvnibkt mc tlby lnvbbquni seiutmek sbts. Xlnt bnfl suffbssmvb systbd ec bquntmeks mk Bxndpib 6.?ms mkhbbh bqumvnibkt te tlb prbvmeus systbd ms `unrnktbbh oy tlb ceiiewmk`tlberbd.

Xlberbd 6.> Xlb systbd ec twe bquntmeks mk k ukakewks evbr n cmbihC


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6.> V]VXBDV EC IMKBN_ B^RNXMEKV >>=

n>>x>+ n

>?x?+!+ n

>kx

k 4 o

>

n?>x>+ n

??x?+!+ n

?kx

k 4 o

?

(>)

wmtl n ": ms bqumvnibkt te tlb systbd

n>>x>+ n

>?x?+!+ n>kxk 4 o>

!n??x?+!+ !n

?kx

k 4 !o

?

(?)

mk wlmfln?m 4 n?> n>m- n>> n?m

cer bnfl m 4 >, . . . , k nkh

o? 4 n?> o>- n>> o? .

Zreec Ibt us hbcmkb

Im 4 nmgg4>

k

! xg

se tlnt (>) dny ob wrmttbk ns tlb systbd

I> 4 o

>

I? 4 o

?

(>)

wlmib (?) ms gustI> 4 o

>

n?>I>! n

>>I

? 4n

?>o>! n

>>o?

(?)

Mc (x, . . . , xk) !Ckms n seiutmek ec (>), tlbk tlb twe bquntmeks

n?>I> 4n

?>o>

n>>I

? 4n

>>o?

nkh lbkfb nisen I - n I 4 n o - n o

nrb nii trub bquntmeks. Xlbrbcerb bvbry seiutmek ec (>) nise sntmscmbs (?).Fekvbrsbiy, suppesb tlnt wb lnvb n seiutmek (x, . . . , x) te tlb systbd

(?). Xlbk fibnriyn I 4 n o

ms n trub bquntmek. Lbkfb, suotrnftmk` tlb sbfekh ec (?) cred tlms `mvbs us


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IMKBN_ B^RNXMEKV NKH DNX_MFBV>?:

n I - (n I - n I) 4 n o - (n o - n o)er

n I 4 n o .

Xlus I 4 o ms nise n trub bquntmek. Xlms slews tlnt nky seiutmek ec (?) ms n

seiutmek ec (>) nise.

Pb pemkt eut tlnt mk tlb preec ec Xlberbd 6.> (ns wbii ns mk Bxndpib 6.?),mt wns ekiy tlb febccmfmbkts tlbdsbivbs tlnt wbrb ec nky hmrbft usb te us. Xlbukakewks x wbrb kbvbr nftuniiy usbh mk nky ec tlb dnkmpuintmeks. Xlms ms tlbrbnsek tlnt wb hbcmkbh tlb dntrmx ec febccmfmbkts (n). Plnt wb kew prefbbhte he ms te `bkbrnimzb tlb noevb dbtleh ec seivmk` systbds ec bquntmeks mk ndnkkbrtlnt utmimzbs tlms dntrmx bxpimfmtiy.

Bxbrfmsbs

>. Cer bnfl ec tlb ceiiewmk` systbds ec bquntmeks, cmkh n seiutmek mc mt bxmsts7

(n) x +?y ! 6z 4 !>

6x ! y+ ?z 4 0

2x + 6y ! 8z 4 ?

(o)

?x +

y ! ?z 4>:

6x + ?y+ ?z 4

>

2x + 8y+ 6z 4 8

(f) x +?y ! 6z 4 3

?x ! y + 8z 4 ?

8x + 6y ! ?z 4>8

?. Hbtbrdmkb wlbtlbr er ket tlb bnfl ec tlb ceiiewmk` twe systbds ms bqumvn-ibkt(evbr tlb fedpibx cmbih)7

(n) x ! y 4: nkh 6x + y 4:

?x + y 4: x + y 4:

(o) !x + y+ 8z 4: nkh x ! z 4:

x + 6y+

/ ?)x + y+ (2 / ?)z 4:

(f) ?x + (!>+ m)y+ t4:

6y ! ?mz+ 2t4:

nkh


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6.? BIBDBKXN_] _EP EZB_NXMEKV >?6

_8!

_?" 8

:

>

> "2 "?

: : ">2 82 02

: : ">? 63 3:

#

$

%%%%

&

'

((((

(!>/>2)_6"

(!>/>?)_8"

> !? ? !> !>8

: > > !2 !?

: : > !6 !2

: : > !6 !2

#

$

%%%%

&

'

((((

Pb sbb tlnt tlb tlmrh nkh ceurtl bquntmeks nrb mhbktmfni, nkh lbkfb wb lnvbtlrbb bquntmeks mk ceur ukakewks7

x

>

! ?x?

+ ?x6

! x

8

4 !>8

x?+ x

6! 2x

8 4 !?

x6! 6x

8 4 !2

Mt ms kew nppnrbkt tlnt tlbrb nrb nk mkcmkmtb kudobr ec seiutmeks obfnusb, mcwb ibt f ! ob nky rbni kudobr, tlbk eur seiutmek sbt ms `mvbk oy x84 f, x646f - 2, x 4 ?f + 6 nkh x 4 -f + ?. #

Xwe d x k dntrmfbs nrb snmh te ob rew bqumvnibkt mc ekb fnk ob trnks-cerdbh mkte tlb etlbr oy n cmkmtb kudobr ec bibdbktnry rew epbrntmeks. Pb

ibnvb mt te tlb rbnhbr te slew tlnt tlms hbcmkbs nk bqumvnibkfb rbintmek ek tlbsbt ec nii d x k dntrmfbs (sbb Bxbrfmsb 6.?.>).

Eur kbxt tlberbd ms ketlmk` derb tlnk n cerdnimzntmek ec nk bnrimbrrbdnra.

Xlberbd 6.? Ibt N nkh O ob tlb nu`dbktbh dntrmfbs ec twe systbds ec dimkbnr bquntmeks mk k ukakewks. Mc N ms rew bqumvnibkt te O, tlbk oetl systbdslnvb tlb sndb seiutmek sbt.

Zreec Mc N ms rew bqumvnibkt te O, tlbk wb fnk `e cred tlb systbd rbprb-

sbktbhoy N te tlb systbd rbprbsbktbh oy O oy n suffbssmek ec tlb epbrntmeks(n), (o) nkh (f) hbsfrmobh noevb. Mt ms fibnr tlnt epbrntmeks (n) nkh (o) lnvb kebccbft ek tlb seiutmeks, nkh tlb dbtleh ec Xlberbd 6.> slews tlnt epbrntmek(f) nise lns ke bccbft.

N dntrmx ms snmh te ob mk rew-bflbiek cerd mc suffbssmvb rews ec tlbdntrmx stnrt eut (cred tlb ibct) wmtl derb nkh derb zbres. Mk pnrtmfuinr, n


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IMKBN_ B^RNXMEKV NKH DNX_MFBV>?8

dntrmx ms snmh te ob mk rbhufbh rew-bflbiek cerd mc mt lns tlb ceiiewmk`prepbrtmbs (wlmfl nrb derb hmccmfuit te stntb prbfmsbiy tlnk tlby nrb te ukhbr-stnkh)7

(>) Nii zbre rews (mc nky) effur obiew nii kekzbre rews.

(?) Xlb cmrst kekzbre bktry (rbnhmk` cred tlb ibct) mk bnfl rew ms bquni te>.

(6) Mc tlb cmrst kekzbre bktry mk tlb mtl rew ms mk tlb gtl feiudk, tlbkbvbry etlbr bktry mk tlb gtlfeiudk ms :.

(8) Mc tlb cmrst kekzbre bktry mk tlb mtlrew ms mk tlb gtlfeiudk, tlbk g 9g 9 ~ ~ ~ .

Pb wmii fnii tlb cmrst (er ibnhmk`) kekzbre bktrmbs mk bnfl rew ec n rew-bflbiekdntrmx tlb hmstmkùmslbh bibdbktsec tlb dntrmx. Xlus, n dntrmx ms mkrbhufbh rew-bflbiek cerd mc tlb hmstmkùmslbh bibdbkts nrb bnfl bquni te >,

nkh tlby nrb tlb ekiy kekzbre bktrmbs mk tlbmr rbspbftmvb feiudks.

Bxndpib 6.2 Xlb dntrmx> ? !6 : >

: : 2 ? !8

: : : 0 6

: : : : :

"

#

$$$$

%

&

''''

ms mk rew-bflbiek cerd out ket mk rbhufbh rew-bflbiek cerd. Lewbvbr, tlb

dntrmx > : 2 : ?

: > ? : 8

: : : > 0

: : : : :

!

"

####

$

%

&&&&

ms mk rbhufbh rew-bflbiek cerd. Ketb tlnt tlb hmstmkùmslbh bibdbkts ec tlbcmrst dntrmx nrb tlb kudobrs >, 2 nkh 0, nkh tlb hmstmkùmslbh bibdbkts ec tlbsbfekh dntrmx nrb tlb kudobrs >, > nkh >. #

Xlb nièrmtld hbtnmibh mk tlb preec ec eur kbxt tlberbd mktrehufbs n tbfl-kmqub`bkbrniiy akewk ns @nussmnk bimdmkntmek.

Xlberbd 6.6 Bvbry d x k dntrmx N ms rew bqumvnibkt te n rbhufbh rew-bflbiekdntrmx.


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IMKBN_ B^RNXMEKV NKH DNX_MFBV>?3

(n)

> !? 6 !>

? !> ? ?

6 > ? 6

"

#

$$$

%

&

''' (o)

> ? !> ? >

? 8 > !? 6

6 3 ? !3 2

"

#

$$$

%

&

'''

(f)

>

6 !>

?: > !2 6

? !2 6 >

8 > > 2

"

#

$$$$

%

&

''''

6. Cer bnfl ec tlb ceiiewmk` systbds, cmkh n seiutmek er slew tlnt ke seiutmekbxmsts7

(n) x + y+ z 4>

?x!

6y+

0z4

: 6x ! ?y+

x+

y+

z4

??x !y+ z 42

(f) x ! y + ?z 48

6x +y + 8z 43

x +y + z 4>

(h) x + 6y+ z 4?

?x + 0y+ 8z 43

x + y ! 8z 4>

(b) x +6y+ z 4:

?x + 0y+ 8z 4:

x + y ! 8z 4:

(c) ?x ! y+ 2z 4>=

x + 2y ! 6z 48

6x

+ ?y+ 8

z42

(`) ?x ! y+ 2z 4>=

x + 2y ! 6z 48

6x + ?y+ 8z 4?2

8. Ibt c, c nkh c6ob bibdbkts ec CQT (m.b., tlb spnfb ec nii rbni-vniubhcukf-tmekshbcmkbh ek ).(n) @mvbk n sbt {x, x, x6} ec rbni kudobrs, hbcmkb tlb 6 x 6 dntrmx C(x) 4(c(x)) wlbrb tlb rews nrb inobiibh oy m nkh tlb feiudks nrb inobiibh oy g.

Zrevb tlnt tlb sbt {c} ms imkbnriy mkhbpbkhbkt mc tlb rews ec tlb dntrmx C(x)nrb imkbnriy mkhbpbkhbkt.(o) Kew nssudb tlnt bnfl c lns cmrst nkh sbfekh hbrmvntmvbs hbcmkbh ek

sedb mktbrvni (n, o) , nkh ibt c(g)hbketb tlb gtlhbrmvntmvb ec c (wlbrb

c(:) ms gust c). Hbcmkb tlb dntrmx P(x) 4 (c (g-> )(x)) wlbrb > m, g 6.Zrevb tlnt {c} ms imkbnriy mkhbpbkhbkt mc tlb rews ec P(x) nrb mkhbpbkhbkt


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6.? BIBDBKXN_] _EP EZB_NXMEKV >?0

cer sedb x !(n, o). (Xlb hbtbrdmknkt ec P(x) ms fniibh tlb Preksamnkectlb sbt ec cukftmeks {cm}.)

Vlew tlnt bnfl ec tlb ceiiewmk` sbts ec cukftmeks ms imkbnriy mkhbpbkhbkt7

(f) c(x) 4 -x?+ x + >, c(x) 4 x?+ ?x, c6(x) 4 x?- >.(h) c(x) 4 bxp(-x), c(x) 4 x, c6(x) 4 bxp(?x).(b) c(x) 4 bxp(x), c(x) 4 smk x, c6(x) 4 fes x.

2. Ibt

N 4

6 !> ?

?

>

>

> !6 :

"

#

$$$

%

&

'''

.

Hbtbrdmkb tlb vniubs ec ] 4 (y, y, y6) cer wlmfl tlb systbd nx 4 y

lns n seiutmek.

3. _bpbnt tlb prbvmeus preoibd wmtl tlb dntrmx

N 4

6 !3 ? !>

!? 8 > 6

: : > >

> !? > :

"

#

$$$$

%

&

''''

.

6.6 _EP NKH FEIRDK VZNFBV

Pb wmii cmkh mt bxtrbdbiy usbcui te feksmhbr tlb rews nkh feiudks ec nk nrom-trnryd x k dntrmx ns vbfters mk tlbmr ewk rm`lt. Mk pnrtmfuinr, tlb rews ec Nnrb te ob vmbwbh ns vbfter k-tupibsN, . . . , Ndwlbrb bnfl N 4 (nm>, . . . ,

nmk) !Ck. Vmdminriy, tlb feiudks ec N nrb te ob vmbwbh ns vbfter d-tupibs

N>, . . . , Nkwlbrb bnfl Ng4 (n>g, . . . , ndg) !Cd. Cer ketntmekni finrmty, wb

sleuih wrmtb Ngns tlb feiudk vbfter

n>g

!

ndg

!

"

###

$

%

&&&

out mt ms type`rnplmfniiy bnsmbr te wrmtb tlms lermzektniiy wlbkbvbr pessmoib.Ketb tlnt wb inobi tlb rew vbfters ec N oy suosfrmpts, nkh tlb feiudks ec Noy supbrsfrmpts.


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IMKBN_ B^RNXMEKV NKH DNX_MFBV>?

d

!

wlbrb bnfl f !C. Xlb spnfb S (wlmfl ms nppnrbktiy n suospnfb ec Ck) msfniibh tlb rew spnfbec N. Xlb hmdbksmek ec S ms fniibh tlb rew rnkaec N,

nkh wmii ob hbketbh oy rr(N). Vmkfb S ms n suospnfb ec Cknkh hmdCk4 k, mtceiiews tlnt rr(N) 4 hmd S k. Ek tlb etlbr lnkh, S ms spnkkbh oy tlb dvbfters N, se tlnt wb dust lnvb hmd S d. Mt tlbk ceiiews tlnt rr(N) dmk{d, k}.

Mk nk bxnftiy nknie`eus dnkkbr, wb hbcmkb tlb feiudk spnfb P ec n

dntrmx N ns tlnt suospnfb ec Cdspnkkbh oy tlb k feiudk vbfters Ng. Xlusnky w !P ms `mvbk oy

w 4 ogNg

g4>

k

!

Xlb feiudk rnkaec N, hbketbh oy fr(N), ms `mvbk oy fr(N) 4 hmd P nkh, nsnoevb, wb dust lnvb fr(N) dmk{d, k}.

Nk eovmeus qubstmek ms wlbtlbr n sbqubkfb ec bibdbktnry rew epbrntmeksflnk`bs bmtlbr tlb rew spnfb er tlb feiudk spnfb ec n dntrmx. N dedbktstleu`lt sleuih fekvmkfb yeu tlnt tlb rew spnfb sleuih ket flnk`b, out mt dnyket ob fibnr bxnftiy wlnt lnppbks te tlb feiudk spnfb. Xlbsb qubstmeks nrbnkswbrbh mk eur kbxt tlberbd. Plmib tlb ceiiewmk` preec nppbnrs te ob rntlbr

iek`, mt ms nftuniiy qumtb smdpib te ukhbrstnkh.

Xlberbd 6.8 Ibt N nkh N ob rew bqumvnibkt d x k dntrmfbs. Xlbk tlb rewspnfb ec N ms bquni te tlb rew spnfb ec N, nkh lbkfb rr(N) 4 rr(N).Curtlbrderb, wb nise lnvb fr(N) 4 fr(N). (Lewbvbr, ketb tlnt tlb feiudkspnfb ec N ms ket kbfbssnrmiy tlb sndb ns tlb feiudk spnfb ec N.)

Zreec Ibt S ob tlb rew spnfb ec N nkh S tlb rew spnfb ec N. Vmkfb N nkh Nnrb rew bqumvnibkt, N dny ob eotnmkbh cred N oy nppiymk` suffbssmvb bib-dbktnry rew epbrntmeks. Out tlbk bnfl rew ec N ms n imkbnr fedomkntmek ec

rews ec N, nkh lbkfb S S. Ek tlb etlbr lnkh, N dny ob eotnmkbh cred Nmk n smdminr dnkkbr se tlnt S S. Xlbrbcerb S 4 S nkh lbkfb rr(N) 4 rr(N).Kew ibt P ob tlb feiudk spnfb ec N nkh P tlb feiudk spnfb ec N.

Rkhbr bibdbktnry rew epbrntmeks, mt wmii ket ob trub mk `bkbrni tlnt P 4 P,out wb wmii slew mt ms stmii niwnys trub tlnt hmd P 4 hmd P. Ibt us hbcmkb tlbdnppmk`c7 P P oy


15/55


16/55

IMKBN_ B^RNXMEKV NKH DNX_MFBV>6:

Pb cmrst feksmhbr n trnkscerdntmek ec typb . Cer hbcmkmtbkbss, wb mktbr-flnk`b rews > nkh ?, nitleu`l mt wmii ob eovmeus tlnt nky pnmr ec rews wmiiwera. Mk etlbr werhs, wb hbcmkb N 4 N, N 4 N nkh N 4 N cer g 4 6, . . . ,k. Xlbrbcerb

c(fNm) 4 fNm4 (fn, fn, fn6, . . . , fnd) .

Mc

fNm 4 :tlbk

fn 4 :

cer bvbry g 4 >, . . . , d nkh lbkfb wb sbb tlnt c(fNm) 4 :. Xlms slews tlnt c mswbii-hbcmkbhcer typb trnkscerdntmeks. Fekvbrsbiy, mc

c(fNm) 4 :tlbk wb sbb tlnt n`nmk

fn 4 :

cer bvbry g 4 >, . . . , d smkfb bnfl fedpekbkt mk tlb bxprbssmek fN m4 :

dust bquni :. Lbkfb fNm4 : mc nkh ekiy mc c(fNm) 4 :, nkh lbkfb Abr c 4{:} cer typb trnkscerdntmeks (wlmfl nise slews tlnt c ms wbii-hbcmkbh).

Pb ibnvb mt te tlb rbnhbr (sbb Bxbrfmsb 6.6.>) te slew tlnt c ms wbii-hbcmkbhnkh Abr c 4 {:} cer trnkscerdntmeks ec typb $, nkh wb `e ek te tlesb ec typb .

N`nmk cer hbcmkmtbkbss, wb feksmhbr tlb pnrtmfuinr trnkscerdntmek N 4 N +aN nkh N 4 N cer g 4 ?, . . . , d. Xlbk

c !fmNm( ) 4 !fm !Nm 4 !fm n>m + an?m ,n?m ,,ndm( )

4 !fmn>m +!afmn?m ,!fmn?m ,,!fmndm( )

Mc

fNm 4 :tlbk

fn 4 :

cer bvbry g 4 >, . . . , d se tlnt fN m4 : nkh c ms wbii-hbcmkbh cer typb trnkscerdntmeks. Fekvbrsbiy, mc

fNm 4 :tlbk

fn 4 :


17/55

6.6 _EP NKH FEIRDK VZNFBV >6>

cer g 4 ?, . . . , d, nkh tlms tlbk slews tlnt fmn>m4 : nise. Xlus fNm4 :

mdpimbs tlnt fNm4 :, nkh lbkfb fNm4 : mc nkh ekiy mc c(fNm) 4 :. Xlmsslews tlnt Abr c 4 {:} cer typb trnkscerdntmeks nise, nkh c ms wbii-hbcmkbh.

Mk suddnry, oy fekstruftmk` nk bxpimfmt msederplmsd mk bnfl fnsb, wblnvb slewk tlnt tlb feiudk spnfbs P nkh P nrb msederplmf ukhbr nii tlrbbtypbs ec bibdbktnry rew epbrntmeks, nkh lbkfb mt ceiiews tlnt tlb feiudkspnfbs ec rew bqumvnibkt dntrmfbs dust lnvb tlb sndb hmdbksmek.

Fereiinry Mc N ms tlb rew-bflbiekcerd ec N, tlbk fNm4 : mc nkh ekiy mc

fNm4 :.

Zreec Xlms wns slewk bxpimfmtiy mk tlb preec ec Xlberbd 6.8 cer bnfl typb ecbibdbktnry rew epbrntmek.

Mk Xlberbd 6.6 wb slewbh tlnt bvbry dntrmx ms rew bqumvnibkt te nrbhufbh rew-bflbiekdntrmx, nkh lbkfb (oy Xlberbd 6.8) nky dntrmx nkh mtsrew fnkekmfni cerd lnvb tlb sndb rew spnfb. Ketb tleu`l, tlnt mc tlb erm`mknidntrmx lns derb rews tlnk tlb hmdbksmek ec mts rew spnfb, tlbk tlb rewseovmeusiy fnk ket nii ob imkbnriy mkhbpbkhbkt. Lewbvbr, wb kew slew tlnt tlbkekzbre rews ec tlb rew fnkekmfni cerd nrb mk cnft imkbnriy mkhbpbkhbkt, nkhlbkfb cerd n onsms cer tlb rew spnfb.

Xlberbd 6.2 Xlb kekzbre rew vbfters ec nk d x k rbhufbh rew-bflbiekdntrmx _ cerd n onsms cer tlb rew spnfb ec _.

Zreec Cred tlb ceur prepbrtmbs ec n rbhufbh rew-bflbiekdntrmx, wb sbb tlntmc _ lns r kekzbre rews, tlbk tlbrb bxmst mktb`brs g, . . . , g rwmtl bnfl g knkh g 9 ~ ~ ~ 9 grsufl tlnt _ lns n > mk tlb mtlrew nkh gtlfeiudk, nkh bvbryetlbr bktry mk tlb gtlfeiudk ms : (mt dny lbip te rbcbr te Bxndpib 6.2 cervmsunimzntmek). Mc wb hbketb tlbsb kekzbre rew vbfters oy _, . . . , _ rtlbk nkynromtrnry vbfter

v 4 fm_

m

m4>

r

!

lns f ns mts gtlfeerhmkntb (ketb tlnt v dny lnvb derb tlnk r feerhmkntbs mc r 9

k). Xlbrbcerb, mc v 4 : wb dust lnvb bnfl feerhmkntb ec v bquni te :, nkhlbkfb f 4 : cer bnfl m 4 >, . . . , r. Out tlms dbnks tlnt tlb _ nrb imkbnriymkhbpbkhbkt, nkh smkfb {_} spnks tlb rew spnfb oy hbcmkmtmek, wb sbb tlnttlby dust mk cnft cerd n onsms.


18/55

IMKBN_ B^RNXMEKV NKH DNX_MFBV>6?

Fereiinry Mc N ms nky dntrmx nkh _ ms n rbhufbh rew-bflbiekdntrmx rewbqumvnibkt te N, tlbk tlb kekzbre rew vbfters ec _ cerd n onsms cer tlb rewspnfb ec N.

Zreec Mk Xlberbd 6.8 wb slewbh tlnt N nkh _ lnvb tlb sndb rew spnfb.

Xlb fereiinry kew ceiiews cred Xlberbd 6.2.

Bxndpib 6.3 Ibt us hbtbrdmkb wlbtlbr er ket tlb ceiiewmk` dntrmfbs lnvbtlb sndb rew spnfb7

N 4

> ? !> 6

? 8 > !?

6 3 6 !0

"

#

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''' O 4

> ? !8 >>

? 8 !2 >8

"

#$

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&' .

Pb ibnvb mt te tlb rbnhbr te slew (nkh yeu rbniiy sleuih he mt) tlnt tlb rbhufbhrew-bflbiekcerd ec tlbsb dntrmfbs ms

N 4

> ? : > / 6

:

:

> !< / 6

: : : :

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O 4> ? : > / 6

:

:

> !< / 6

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Vmkfb tlb tlb kekzbre rews ec tlb rbhufbh rew-bflbiek cerd ec N nkh O nrbmhbktmfni, tlb rew spnfbs dust ob tlb sndb. #

Kew tlnt wb lnvb n obttbr ukhbrstnkhmk` ec tlb rew spnfb ec n dntrmx, ibtus è onfa nkh slew tlnt tlb rbhufbh rew-bflbiekcerd ec n `mvbk dntrmx msukmqub. Pb cmrst prevb n prbimdmknry rbsuit hbnimk` wmtl tlb rew-bflbiekcerdec twe dntrmfbs lnvmk` tlb sndb rew spnfb.

Xlberbd 6.3 Ibt N 4 (n) ob n rew-bflbiekdntrmx wmtl hmstmkùmslbh bib-dbktsn>g, n?g, . . . , nrg nkh ibt O 4 (o) ob nketlbr rew-bflbiekdntrmx wmtlhmstmkùmslbh bibdbkts o>a, o?a, . . . , osa . Nssudb tlnt N nkh O lnvb tlbsndb rew spnfb (nkh tlbrbcerb tlb sndb kudobr ec feiudks). Xlbk tlb hms-tmkùmslbhbibdbkts ec N nrb mk tlb sndb pesmtmek ns tlesb ec O, m.b., r 4 s nkh

g 4 a, g 4 a, . . . , gr4 ar .

Zreec Vmkfb N 4 : mc nkh ekiy mc O 4 :, wb kbbh ekiy feksmhbr tlb kektrmvmnifnsb wlbrb r > nkh s >. Cmrst suppesb tlnt g 9 a. Xlms dbnks tlnt tlb g tlfeiudk ec O ms bquni te (:, . . . , :). Vmkfb N nkh O lnvb tlb sndb rew spnfb,tlb cmrst rew ec N ms gust n imkbnr fedomkntmek ec tlb rews ec O. Mk pnrtmfuinr,wb tlbk lnvb n>g 4 fomg cer sedb sbt ec sfninrs f (sbb tlb preec ec


19/55

6.6 _EP NKH FEIRDK VZNFBV >66

Xlberbd 6.8). Out omg4 : cer bvbry m, nkh lbkfb n>g4 : wlmfl fektrnhmfts tlbnssudptmektlnt n>gms n hmstmk`umslbh bibdbkt ec N (nkh dust ob kekzbre oyhbcmkmtmek). Pb nrb tlus cerfbh te fekfiuhb tlnt g a. Lewbvbr, wb feuihfibnriy lnvb stnrtbh wmtl tlb nssudptmek tlnt a 9 g, mk wlmfl fnsb wb weuihlnvb obbk ibh te fekfiuhb tlnt a g. Xlms slews tlnt wb dust nftuniiy lnvb

g 4 a.Kew ibt N nkh O ob tlb dntrmfbs wlmfl rbsuit cred hbibtmk` tlb cmrst rew

ec N nkh O rbspbftmvbiy. Mc wb fnk slew tlnt N nkh O lnvb tlb sndb rewspnfb, tlbk tlby wmii nise sntmscy tlb lypetlbsbs ec tlb tlberbd, nkh eur fek-fiusmekceiiews nt ekfb oy mkhuftmek.

Ibt _ 4 (n, n, . . . , n) ob nky rew ec N (nkh lbkfb n rew ec N), nkh ibtO, . . . , Odob tlb rews ec O. Vmkfb N nkh O lnvb tlb sndb rew spnfb, wbn`nmk lnvb

_ 4 hmOm

m4>

d

!

cer sedb sbt ec sfninrs h. Vmkfb _ ms ket tlb cmrst rew ec N nkh N ms mk rew-bflbiekcerd, mt ceiiews tlnt n 4 : cer m 4 g 4 a. Mk nhhmtmek, tlb cnft tlnt O msmk rew-bflbiekcerd dbnks tlnt bvbry bktry mk tlb atlfeiudk ec O dust ob :bxfbpt cer tlb cmrst, m.b., o>a":, o?a4 ~ ~ ~ 4 oda4 :. Out tlbk

: 4 na 4 h o>a+ h o?a+ ~ ~ ~ + hd oda 4 h o>a

wlmfl mdpimbs tlnt h 4 : smkfb o>a":. Xlms slews tlnt _ ms nftuniiy n imkbnrfedomkntmek ec tlb rews ec O, nkh lbkfb (smkfb _ wns nromtrnry) tlb rew

spnfb ec N dust ob n suospnfb ec tlb rew spnfb ec O. Xlms nr`udbkt fnkfibnriy ob rbpbntbh te slew tlnt tlb rew spnfb ec O ms n suospnfb ec tlb rewspnfb ec N, nkh lbkfb wb lnvb slewk tlnt N nkh O lnvb tlb sndb rewspnfb.

Xlberbd 6.0 Ibt N 4 (n) nkh O 4 (o) ob rbhufbh rew-bflbiekdntrmfbs.Xlbk N nkh O lnvb tlb sndb rew spnfb mc nkh ekiy mc tlby lnvb tlb sndbkekzbre rews.

Zreec Vmkfb mt ms eovmeus tlnt N nkh O lnvb tlb sndb rew spnfb mc tlby lnvbtlb sndb kekzbre rews, wb kbbh ekiy prevb tlb fekvbrsb. Ve, suppesb tlnt Nnkh O lnvb tlb sndb rew spnfb. Xlbk mc N ms nk nromtrnry kekzbre rew ec N,wb dny wrmtb

Nm 4 !

rfrO

r (>)

wlbrb tlb Ornrb tlb kekzbre rews ec O. Xlb preec wmii ob cmkmslbh mc wb fnkslew tlnt fr4 : cer r "m nkh f 4 >.


20/55

IMKBN_ B^RNXMEKV NKH DNX_MFBV>68

Xe slew tlnt f 4 >, ibt nmgob tlb cmrst kekzbre bktry mk N, m.b., nmgms tlbhmstmkùmslbh bibdbkt ec tlb mtl rew ec N. Ieeamk` nt tlb gtlfedpekbkt ec(>) wb sbb tlnt

n mgm 4 !rfrorgm (?)

(sbb tlb preec ec Xlberbd 6.8). Cred Xlberbd 6.3 wb akew tlnt omg ms tlbhmstmkùmslbh bibdbkt ec tlb mtl rew ec O, nkh lbkfb mt ms tlb ekiy kekzbrebktry mk tlb gtl feiudk ec O (oy hbcmkmtmek ec n rbhufbh rew-bflbiekdntrmx).Xlms dbnks tlnt (?) mdpimbs nmg4 f omg. Mk cnft, mt dust ob trub tlnt nmg4 omg4> smkfb N nkh O nrb rbhufbh rew-bflbiekdntrmfbs, nkh tlbrbcerb f 4 >.

Kew ibt oagob tlb cmrst kekzbre bktry ec O (wlbrb a "m). Cred (>) n`nmkwb lnvb

nmga 4 !rfrorga . (6)

Vmkfb O ms n rbhufbh rew-bflbiekdntrmx, oag4 > ms tlb ekiy kekzbre bktry mktlb gtl feiudk ec O, nkh lbkfb (6) slews us tlnt nmg 4 foag . Out credXlberbd 6.3, nagms n hmstmkùmslbh bibdbkt ec N, nkh lbkfb tlb cnft tlnt N msrew-rbhufbhdbnks tlnt nmg4 : cer m "a. Xlms cerfbs us te fekfiuhb tlnt f 4: cer a "m ns finmdbh.

Vuppesb tlnt twe pbepib nrb `mvbk tlb sndb dntrmx N nkh nsabh te trnks-cerdmt te rbhufbh rew-bflbiekcerd _. Xlb flnkfbs nrb qumtb èeh tlnt tlbywmii bnfl pbrcerd n hmccbrbkt sbqubkfb ec bibdbktnry rew epbrntmeks tenflmbvb tlb hbsmrbh rbsuit. Ibt _ nkh _ ob tlb rbhufbh rew-bflbiekdntrmfbs

tlnt eur twe stuhbkts eotnmk. Pb finmd tlnt _ 4 _. Mkhbbh, smkfb rew bqumvn-ibkfbhbcmkbsnk bqumvnibkfb rbintmek, wb sbb cred Xlberbd 6.8 tlnt tlb rewspnfbs ec _ nkh _ wmii ob tlb sndb. Xlbrbcerb Xlberbd 6.0 slews us tlnt tlbrews ec _ dust bquni tlb rews ec _. Lbkfb wb nrb gustmcmbh mk fniimk` tlbrbhufbh rew-bflbiek cerd ec n dntrmx tlbrew fnkekmfni cerd ns dbktmekbhbnrimbr.

Bxbrfmsbs

>. Mk tlb preec ec Xlberbd 6.8, slew tlnt Abr c 4 {:} cer n typb $epbrntmek.

?. Hbtbrdmkb wlbtlbr er ket tlb ceiiewmk` dntrmfbs lnvb tlb sndb rewspnfb7

N 4> !? !>

6 !8 2

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> !> ?

6 6 !>

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> !> 6

? !> >:

6 !2 >

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21/55

6.6 _EP NKH FEIRDK VZNFBV >62

6. Vlew tlnt tlb suospnfb ec 6spnkkbh oy tlb vbfters u 4 (>, >, ->), u 4(?, 6, ->) nkh u64 (6, >, -2) ms tlb sndb ns tlb suospnfb spnkkbh oy tlbvbfters v 4 (>, ->, -6), v 4 (6, -?, -, -6).

8. Hbtbrdmkb wlbtlbr er ket bnfl ec tlb ceiiewmk` sbts ec vbfters ms imkbnriymkhbpbkhbkt7(n) u 4 (>, -?, >), u 4 (?, >, ->) nkh u64 (0, -8, >).(o) u 4 (>, ?, -6), u 4 (>, -6, ?) nkh u64 (?, ->, 2).

2. (n) Vuppesb wb nrb `mvbk nk d x k dntrmx N 4 (n), nkh suppesb tlnt ekb

ec tlb feiudks ec N, sny Nm, ms n imkbnr fedomkntmek ec tlb etlbrs. Vlewtlnt ukhbr nky bibdbktnry rew epbrntmek rbsuitmk` mk n kbw dntrmx N, tlb

feiudk Nmms tlb sndb imkbnr fedomkntmek ec tlb feiudks ec N tlnt Nmms ec

tlb feiudks ec N. Mk etlbr werhs, slew tlnt nii imkbnr rbintmeks obtwbbkfeiudks nrb prbsbrvbh oy bibdbktnry rew epbrntmeks.(o) Rsb tlms rbsuit te `mvb nketlbr preec ec Xlberbd 6.8.(f) Rsb tlms rbsuit te `mvb nketlbr preec ec Xlberbd 6.0.

6.8 XLB _NKA EC N DNX_M\

Mt ms mdpertnkt cer tlb rbnhbr te rbnimzb tlnt tlbrb ms ketlmk` spbfmni noeut tlbrews ec n dntrmx. Bvbrytlmk` tlnt wb lnvb hekb up te tlms pemkt mk hmsfussmk`bibdbktnry rew epbrntmeks feuih gust ns bnsmiy lnvb obbk hekb wmtl feiudksmkstbnh. Mk pnrtmfuinr, tlms dbnks tlnt Xlberbds 6.8 nkh 6.2 nrb bquniiy vnimhcer feiudk spnfbs mc wb nppiy eur bibdbktnry trnkscerdntmeks te feiudksmkstbnh ec rews. Xlms eosbrvntmek ibnhs us te eur kbxt cukhndbktni rbsuit.

Xlberbd 6.


22/55


r(N) 4 rr(N) 4 fr(N) .

Eur kbxt tlberbd cerds tlb onsms cer n prnftmfni dbtleh ec cmkhmk` tlb rnka ecn dntrmx.

Xlberbd 6.= Mc N ms nky dntrmx, tlbk r(N) ms bquni te tlb kudobr ec kekzbrerews mk tlb (rbhufbh) rew-bflbiekdntrmx rew bqumvnibkt te N. (Nitbrkntmvbiy,r(N) ms tlb kudobr ec kekzbre feiudks mk tlb (rbhufbh) feiudk-bflbiekdntrmx feiudk bqumvnibkt te N.)

Zreec Ketmk` tlnt tlb kudobr ec kekzbre rews mk tlb rew-bflbiekcerd ms tlbsndb ns tlb kudobr ec kekzbre rews mk tlb rbhufbh rew-bflbiekcerd, wb sbbtlnt tlms tlberbd ceiiews hmrbftiy cred tlb fereiinry te Xlberbd 6.2.

Mc N ms nk k x k dntrmx sufl tlnt n 4 : cer m "g nkh n 4 >, tlbk wb sny tlnt

N ms tlb mhbktmty dntrmx

ec smzb k nkh wrmtb tlms dntrmx ns M. Vmkfb tlb smzb msusuniiy ukhbrsteeh, wb wmii `bkbrniiy smdpiy wrmtb M. Mc M 4 (M), tlbk nketlbrusbcui wny ec wrmtmk` tlms ms mk tbrds ec tlb Arekbfabr hbitn ns M 4 %.Prmttbk eut, M lns tlb cerd

M 4

> : : ! :

: > : ! :

" " " "

: : : ! >

!

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####

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%

&&&&

.

Xlberbd 6.>: Mc N ms nk k x k dntrmx ec rnka k, tlbk tlb rbhufbh rew-bflbiekdntrmx rew bqumvnibkt te N ms tlb mhbktmty dntrmx M.

Zreec Xlms ceiiews cred tlb hbcmkmtmek ec n rbhufbh rew-bflbiekdntrmx nkhXlberbd 6.=.

Bxndpib 6.0 Ibt us cmkh tlb rnka ec tlb dntrmx N `mvbk oy

N 4

>

? !6

? > :

!? !> 6

!> 8 !?

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.


23/55


24/55

IMKBN_ B^RNXMEKV NKH DNX_MFBV>6. Sbrmcy tlb rew-fnkekmfnicerd ec tlb dntrmx mk Bxndpib 6.0.

?. Ibt N nkh O ob nromtrnry d x k dntrmfbs. Vlew tlnt r(N + O) r(N) + r(O).

6. Rsmk` bibdbktnry rew epbrntmeks, cmkh tlb rnka ec bnfl ec tlb ceiiewmk`dntrmfbs7

(n)

> 6 > !? !6

> 8 6 !> !8

?

6 !8 !0 !6

6 < > !0 !

6 : !?

2 !>

!? 6

"

#

$$$$

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(h)

2 !>

>

? > !?

: !0 >?

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8. _bpbnt tlb prbvmeus preoibd usmk` bibdbktnry feiudk epbrntmeks.

6.2 VEIRXMEKV XE V]VXBDV EC IMKBN_ B^RNXMEKV

Pb kew nppiy tlb rbsuits ec tlb prbvmeus sbftmek te tlb hbtbrdmkntmek ec sedb`bkbrni flnrnftbrmstmfs ec tlb seiutmek sbt te systbds ec imkbnr bquntmeks. Pbwmii lnvb derb te sny ek tlms suogbft nctbr wb lnvb hmsfussbh hbtbrdmknkts mktlb kbxt flnptbr.

Xe ob`mk wmtl, n systbd ec imkbnr bquntmeks ec tlb cerd

nmgxgg4>

k

! 4:, m 4>,,d

ms fniibh n lede`bkbeus systbdec d imkbnr bquntmeks mk k ukakewks. Mt mseovmeus tlnt fleesmk` x 4 x 4 ~ ~ ~ 4 x 4 : wmii sntmscy tlms systbd, out tlms

ms ket n vbry mktbrbstmk` seiutmek. Mt ms fniibh tlb trmvmni (er zbre) seiutmek.Nky etlbr seiutmek, mc mt bxmsts, ms rbcbrrbh te ns n kektrmvmni seiutmek.N derb `bkbrni typb ec systbd ec imkbnr bquntmeks ms ec tlb cerd

nmgxg 4ym , m 4>,,d

g4>

k

!

wlbrb bnfl y ms n `mvbk sfninr. Xlms ms tlbk fniibh n keklede`bkbeus systbdec imkbnr bquntmeks. Ibt us hbcmkb tlb feiudk vbfter


25/55

6.2 VEIRXMEKV XE V]VXBDV EC IMKBN_ B^RNXMEKV >6=

] 4 (y, . . . , yd) !Cd .

Cred eur hmsfussmek mk tlb preec ec Xlberbd 6.8, wb sbb tlnt nx ms gust x

tmdbs tlb mtlfedpekbkt ec tlb gtlfeiudk Ng!Cd. Xlus eur systbd ec kek-lede`bkbeus

bquntmeks dny ob wrmttbk mk tlb cerd

Ngxg 4x>

n>>

n?>

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nd>

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+"+xk

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k

'

wlbrb tlms vbfter bquntmek ms te ob mktbrprbtbh mk tbrds ec mts fedpekbkts. (Mktlb kbxt sbftmek, wb slnii sbb lew te wrmtb tlms ns n prehuft ec dntrmfbs.) Mtsleuih nise ob eovmeus tlnt n lede`bkbeus systbd dny ob wrmttbk mk tlms

ketntmek ns

Ngxg 4: .

g4>

k

!

Ibt us kew ieea nt sedb bibdbktnry prepbrtmbs ec tlb seiutmek sbt ec nlede`bkbeussystbd ec bquntmeks.

Xlberbd 6.>> Xlb seiutmek sbt V ec n lede`bkbeus systbd ec d bquntmeks

mk k ukakewks ms n suospnfb ecCk.

Zreec Ibt us wrmtb eur systbd ns gnx 4 :. Pb cmrst ketb tlnt V " smkfb(:, . . . , :) !Ckms tlb trmvmni seiutmek ec eur systbd. Mc u 4 (u, . . . , u) !Ck

nkh v 4 (v, . . . , v) !Cknrb oetl seiutmeks, tlbk

n(u + v) 4 nu + nv 4 :

se tlnt u + v !V. Cmkniiy, mc f !Ctlbk wb nise lnvb

n(fu) 4 fnu 4 :se tlnt fu ! V.

Mc wb ieea onfa nt Bxndpib 6.8, wb sbb tlnt n systbd ec d bquntmeks mk k ;d ukakewks wmii kbfbssnrmiy rbsuit mk n kekukmqub, nkh lbkfb kektrmvmni, seiu-tmek. Xlb cerdni stntbdbkt ec tlms cnft ms fektnmkbh mk eur kbxt tlberbd.


26/55


Xlberbd 6.>? Ibt n lede`bkbeus systbd ec d bquntmeks mk k ukakewkslnvb tlb d x k dntrmx ec febccmfmbkts N. Xlbk tlb systbd lns n kektrmvmniseiutmek mc nkh ekiy mc r(N) 9 k.

Zreec Oy wrmtmk` tlb systbd mk tlb cerd xNg4 :, mt ms fibnr tlnt n kek-

trmvmniseiutmek bxmsts mc nkh ekiy mc tlb k feiudk vbfters Ng!Cdnrb imkbnriyhbpbkhbkt. Vmkfb tlb rnka ec N ms bquni te tlb hmdbksmek ec mts feiudk spnfb,wb dust tlbrbcerb lnvb r(N) 9 k.

Mt sleuih kew ob fibnr tlnt mc nk k x k (m.b., squnrb) dntrmx ec febccmfmbktsN (ec n lede`bkbeus systbd) lns rnka bquni te k, tlbk tlb ekiy seiutmek wmiiob tlb trmvmni seiutmek smkfb rbhufmk` tlb nu`dbktbh dntrmx (wlmfl tlbk lnstlb inst feiudk bquni te tlb zbre vbfter) te rbhufbh rew-bflbiek cerd wmiirbsuit mk bnfl vnrmnoib obmk` sbt bquni te zbre (sbb Xlberbd 6.>:).

Xlberbd 6.>6 Ibt n lede`bkbeus systbd ec imkbnr bquntmeks mk k ukakewkslnvb n dntrmx ec febccmfmbkts N. Xlbk tlb seiutmek sbt V ec tlms systbd ms n suo-

spnfbecCkwmtl hmdbksmek k - r(N).

Zreec Nssudb tlnt V ms n kektrmvmni seiutmek sbt, se tlnt oy Xlberbd 6.>? wblnvb r(N) 9 k. Nssudb nise tlnt tlb ukakewks x, . . . , x lnvb obbk erhbrbhmk sufl n wny tlnt tlb cmrst a 4 r(N) feiudks ec N spnk tlb feiudk spnfb (tlms

ms `unrnktbbh oy Xlberbd 6.8). Xlbk tlb rbdnmkmk` feiudks Na+>, . . . , Nkdny ob wrmttbk ns

Nm 4 omgNg

g4>

a

! , m 4a+>,,k

nkh wlbrb bnfl o !C. Mc wb hbcmkb o 4 -> nkh o 4 : cer g "m nkh g ; a,tlbk wb dny wrmtb tlms ns

omgNg4:, m 4a+>,,k

g4>

k

!

(ketb tlb uppbr imdmt ek tlms sud hmccbrs cred tlb prbvmeus bquntmek). Kbxt wb

eosbrvb tlnt tlb seiutmek sbt V feksmsts ec nii vbfters x !Cksufl tlnt

xgNg 4:

g4>

k

! nkh lbkfb mk pnrtmfuinr, tlb k - a vbfters

o(m) 4 (o, . . . , o)


27/55

6.2 VEIRXMEKV XE V]VXBDV EC IMKBN_ B^RNXMEKV >8>

cer bnfl m 4 a + >, . . . , k dust obiek` te V. Pb slew tlnt tlby mk cnft cerd nonsms cer V, wlmfl ms tlbk ec hmdbksmek k - a.

Xe sbb tlms, wb cmrst wrmtb eut bnfl ec tlb o(m)7

o(a+>) 4 (oa+>>,,oa+>a,!>,:,:,,:)

o(a+?) 4 (oa+? >,,oa+? a,:,!>,:,,:)

!

o(k) 4 (ok>,,oka,:,:,,:,!>) .

Lbkfb cer nky sbt {f} ec k - a sfninrs wb lnvb

fmo(m)

m4a+>

k

! 4 fmom>,, fmomk,"fa+>,,"fkm4a+>

k

!m4a+>

k

!#

$%%

&

'((

nkh tlbrbcerb

fmo(m) 4:

m4a+>

k

!

mc nkh ekiy mc fa+>4 ~ ~ ~ 4 f 4 :. Xlms slews tlnt tlb o(m) nrb imkbnriy mkhb-pbkhbkt. (Xlms sleuih lnvb obbk eovmeus cred tlbmr cerd slewk noevb.)

Kew suppesb tlnt h 4 (h, . . . , h) ms nky seiutmek ec

xgNg4: .

g4>

k

!

Vmkfb V ms n vbfter spnfb (Xlberbd 6.>>), nky imkbnr fedomkntmek ec seiutmeksms n seiutmek, nkh lbkfb tlb vbfter

y 4h+ hmo(m)

m4a+>

k

!

dust nise ob n seiutmek. Mk pnrtmfuinr, wrmtmk` eut bnfl fedpekbkt ec tlmsbxprbssmekslews tlnt

yg 4hg + hmomgm4a+>

k

!

nkh lbkfb tlb hbcmkmtmek ec tlb o slews tlnt y 4 (y, . . . , y, :, . . . , :) cersedb sbt ec sfninrs y. Xlbrbcerb, wb lnvb

: 4 ygNg

4 ygNg

g4>

a

!g4>

k

!

nkh smkfb {N>, . . . , Na} ms imkbnriy mkhbpbkhbkt, tlms mdpimbs tlnt y 4 : cerbnfl g 4 >, . . . , a. Lbkfb y 4 : se tlnt

h4 ! hmo(m)

m4a+>

k

"


28/55


29/55

6.2 VEIRXMEKV XE V]VXBDV EC IMKBN_ B^RNXMEKV >86

Bxndpib 6., w 4 :, t 4 :),

(y 4 :, w 4 >, t 4 :) nkh (y 4 :, w 4 :, t 4 >). Rsmk` bnfl ec tlb tlbsb mk tlbsystbd (*), wb eotnmk tlb seiutmeks

v> 4 (!?,>,:,:,:)

v? 4 (!>,:,> / ?,>,:)

v6 4 (!6,:,!>,:,>)

Xlus tlb vbfters v, v nkh v6cerd n onsms cer tlb seiutmek spnfb ec tlb lede-`bkbeussystbd. #

Pb bdplnsmzb tlnt tlb fereiinry te Xlberbd 6.8 slews us tlnt tlb seiutmeksbt ec n lede`bkbeus systbd ec bquntmeks ms ukflnk`bh oy bibdbktnry rewepbrntmeks. Mt ms tlms cnft tlnt niiews us te prefbbh ns wb hmh mk Bxndpib 6.8 Ibt n keklede`bkbeus systbd ec imkbnr bquntmeks lnvbdntrmx ec febccmfmbkts N. Xlbk tlb systbd lns n seiutmek mc nkh ekiy mc r(N) 4r(nu` N).

Zreec Ibt f 4 (f, . . . , f) ob n seiutmek ec nx 4 y. Xlbk wrmtmk` tlms ns

fNg 4 ]

slews us tlnt ] ms mk tlb feiudk spnfb ec N, nkh lbkfb

r(nu` N) 4 fr(nu` N) 4 fr(N) 4 r(N) .


30/55


Fekvbrsbiy, mc fr(nu` N) 4 r(nu` N) 4 r(N) 4 fr(N), tlbk ] ms mk tlb feiudk

spnfb ec N, nkh lbkfb ] 4 fNg cer sedb sbt ec sfninrs f . Out tlbk tlbvbfter f 4 (f, . . . , f) ms n seiutmek smkfb mt eovmeusiy sntmscmbs nx 4 y.

Rsmk` Xlberbd 6.>6, mt ms bnsy te hbsfrmob tlb `bkbrni seiutmek te n kek-lede`bkbeussystbd ec bquntmeks.

Xlberbd 6.>2 Ibt

nmgxg 4ygg4>

k

!

ob n systbd ec keklede`bkbeus imkbnr bquntmeks. Mc u 4 (u, . . . , u) !Ckms nseiutmek ec tlms systbd, nkh mc V ms tlb seiutmek spnfb ec tlb nssefmntbh lede-`bkbeussystbd, tlbk tlb sbt

u + V 4 {u + v7 v !V}

ms tlb seiutmek sbt ec tlb keklede`bkbeus systbd.

Zreec Mc w 4 (w, . . . , w)!Ckms nky etlbr seiutmek ec nx 4 y, tlbk

n(w - u) 4 nw - nu 4 y - y 4 :

se tlnt w - u !V, nkh lbkfb w 4 u + v cer sedb v !V. Fekvbrsbiy, mc v !V

tlbkn(u + v) 4 nu + nv 4 y + : 4 y

se tlnt u + v ms n seiutmek ec tlb keklede`bkbeus systbd.

Xlberbd 6.>3 Ibt N ob nk k x k dntrmx ec rnka k. Xlbk tlb systbd

Ngxg 4]

g4>

k

!

lns n ukmqub seiutmek cer nromtrnry vbfters ] !Ck.

Zreec Vmkfb ] 4 Ngx, wb sbb tlnt ] !Ckms gust n imkbnr fedomkntmek ectlb feiudks ec N. Vmkfb r(N) 4 k, mt ceiiews tlnt tlb feiudks ec N nrb imkbnriy

mkhbpbkhbkt nkh lbkfb cerd n onsms cerCk. Out tlbk nky ] !Cklns n ukmqubbxpnksmek mk tbrds ec tlms onsms (Xlberbd ?.8, Fereiinry ?) se tlnt tlb vbfter\ wmtl fedpekbkts x dust ob ukmqub.


31/55


Bxndpib 6.= Ibt us cmkh tlb fedpibtb seiutmek sbt evbr tlb rbni kudobrs ectlb keklede`bkbeus systbd

6x>+

x?+

?x6+

8x8 4

>

x>! x

?+ 6x

6! x

8 4 6

x>+ 0x

?!>>x

6+>6x

8 4 !>6

>>x>+

x?+>?x

6+>:x

8 4

=

Pb nssudb tlnt wb sedblew ceukh n pnrtmfuinr seiutmek u 4 (?, 2, >, -6) !

8, nkh lbkfb wb sbba tlb seiutmek sbt V ec tlb nssefmntbh lede`bkbeus sys-tbd. Xlb dntrmx ec febccmfmbkts N ec tlb lede`bkbeus systbd ms `mvbk oy

N 4

6 > ? 8

> !> 6 !>

>

0 !>> >6

>>

>

>? >:

"

#

$$$$

%

&

''''

.

Xlb cmrst tlmk` wb dust he ms hbtbrdmkb r(N). Vmkfb tlb preec ec Xlberbd 6.>6hbnit wmtl feiudks, wb fleesb te fekstruft n kbw dntrmx O oy nppiymk` bib-dbktnryfeiudk epbrntmeks te N. Xlus wb hbcmkb

O 4

> : : :

!> 8 2 6

0 !?: !?2 !>2 > < >: 3

"

#

$$

$$

%

&

''

''

wlbrb tlb feiudks ec O nrb `mvbk mk tbrds ec tlesb ec N oy O >4 N?, O?4

N> - 6N?, O64 N6 - ?N?nkh O84 N8 - 8N?. Mt ms eovmeus tlnt O>nkh O?nrb

mkhbpbkhbkt, nkh wb nise ketb tlnt O6 4 (2/8)O? nkh O8 4 (6/8)O?. Xlbkr(N) 4 r(O) 4 ?, nkh lbkfb wb lnvb hmd V 4 8 - ? 4 ?.

(Nk nitbrkntmvb dbtleh ec cmkhmk` r(N) ms ns ceiiews. Mc wb mktbrflnk`b tlbcmrst twe rews ec N nkh tlbk nhh n sumtnoib duitmpib tlb kbw cmrst rew te bimdm-

kntbtlb cmrst bktry mk bnfl ec tlb rbdnmkmk` tlrbb rews, wb eotnmk

> !> 6 !>

: 8 !0 0

: < !>8 >8

: >? !?> ?>

"

#

$$$$

%

&

''''

.


32/55


Mt ms kew fibnr tlnt tlb cmrst twe rews ec tlms dntrmx nrb mkhbpbkhbkt, nkh tlnttlb tlmrh nkh ceurtl rews nrb bnfl duitmpibs ec tlb sbfekh. Xlbrbcerb r(N) 4 ?ns noevb.)

Pb kew ceiiew tlb cmrst pnrt ec tlb preec ec Xlberbd 6.>6. Eosbrvb tlntsmkfb r(N) 4 ? nkh tlb cmrst twe feiudks ec N nrb mkhbpbkhbkt, wb dny wrmtb

N6 4 (2/8)N> - (0/8)N?nkh

N8 4 (6/8)N>+ (0/8)N? .

Pb tlbrbcerb hbcmkb tlb vbfters

o(6) 4 (2/8, -0/8, ->, :)nkh

o(8)

4 (6/8, 0/8, :, ->)wlmfl nrb mkhbpbkhbkt seiutmeks ec tlb lede`bkbeus systbd nkh spnk tlbseiutmek spnfb V. Xlbrbcerb tlb `bkbrni seiutmek sbt te tlb keklede`bkbeussystbd ms `mvbk oy

u + V4 {u +!o(6) + "o(8 )}

4 {(?,2,>,#6)+!(2 / 8,#0 / 8,#>,:) + "(6 / 8,0 / 8,:,>)}

wlbrb , $! nrb nromtrnry. #

Bxbrfmsbs

>. Cmkh tlb hmdbksmek nkh n onsms cer tlb seiutmek spnfb ec bnfl ec tlb cei-iewmk`systbds ec imkbnr bquntmeks evbr 7

(n) x + 8y+ ?z 4:

?x + y+ 2z 4:

(o) x + 6y+?z 4:

x + 2y+ z 4:

6x + 2y+


33/55


?. Feksmhbr tlb suospnfbs R nkh S ec 8`mvbk oy

R 4 {(n,o,f,h)! !8 7o+ f+ h4:}

S 4 {(n,o,f,h)! !8 7n+o 4:nkh f 4?h} .

(n) Cmkh tlb hmdbksmek nkh n onsms cer R.(o) Cmkh tlb hmdbksmek nkh n onsms cer S.(f) Cmkh tlb hmdbksmek nkh n onsms cer R S.

6. Cmkh tlb fedpibtb seiutmek sbt ec bnfl ec tlb ceiiewmk` systbds ec imkbnrbquntmeks evbr 7

(n) 6x ! y 40

?x + y 4>

(o) ?x ! y+ 6z 4 2

6x + ?y ! ?z 4 >

0x +

8z 4>>

(f) 2x + ?y ! z 4:

6x + 2y+ 6z 4:

x +

6x + y+ z ! 6w 4?

6x ! ?y + 3z 40

6.3 DNX_M\ NI@BO_N

Pb kew mktrehufb tlb bibdbktnry ni`bornmf epbrntmeks ek dntrmfbs. Xlbsbepbrntmeks wmii ob ec tlb utdest mdpertnkfb tlreu`leut tlb rbdnmkhbr ec tlmstbxt. Mk Flnptbr 2 wb wmii sbb lew tlbsb hbcmkmtmeks nrmsb mk n knturni wnycred tlb ni`born ec imkbnr trnkscerdntmeks.

@mvbk twe d x k dntrmfbs N 4 (n) nkh O 4 (o), wb hbcmkb tlbmr sudN +O te ob tlb dntrmx wmtl bktrmbs

(N + O) 4 n + o

eotnmkbh oy nhhmk` tlb ferrbspekhmk` bktrmbs ec bnfl dntrmx. Ketb tlnt oetl N

nkh O dust ob ec tlb sndb smzb. Pb nise sny tlnt N bqunisO mc n 4 o cer niim nkh g. Mt ms eovmeus tlnt

N + O 4 O + Nnkh tlnt

N + (O + F) 4 (N + O) + F


34/55


k

!m4>

d

! wlbrb tlb d x k dntrmx B ms hbcmkbh ns lnvmk` n > mk tlb (m, g) tlpesmtmek nkh:s bisbwlbrb, nkh tlbrb nrb fibnriy dk sufl dntrmfbs. Pb hbketb tlb spnfb ecnii d x k dntrmfbs evbr tlb cmbihCoy Ddxk(C). Xlb pnrtmfuinr fnsb ec d 4 khbcmkbs tlb spnfb Dk(C) ec nii squnrbdntrmfbs ec smzb k. Pb wmii ectbk rbcbrte n dntrmx mk Dk(C) ns nk k-squnrbdntrmx.

Kew ibt N !Ddxk(C) ob nk d x k dntrmx, O ! Drxd(C) ob nk r x ddntrmx, nkh feksmhbr tlb twe systbds ec imkbnr bquntmeks

nmgxg 4ym , m 4>,,d

g4>

k

!

nkh

omgyg 4zm , m 4>,,r

g4>

d

!

wlbrb \ 4 (x, . . . , x) !Ck, ] 4 (y, . . . , yd) !Cdnkh W 4 (z, . . . , zr) !

Cr. Vuostmtutmk` tlb cmrst ec tlbsb bquntmeks mkte tlb sbfekh ymbihs

z 4 oy 4 onx 4 fx

wlbrb wb hbcmkbh tlb prehuftec tlb r x d dntrmx O nkh tlb d x k dntrmx N teob tlb r x k dntrmx F 4 ON wlesb bktrmbs nrb `mvbk oy

fma 4 omgnga .

g4>

d

!

Xlus tlb (m, a)tlbktry ec F 4 ON ms `mvbk oy tlb stnkhnrh sfninr prehuft


35/55

6.3 DNX_M\ NI@BO_N >8=

(ON) 4 O Na

ec tlb mtlrew ec O wmtl tlb atlfeiudk ec N (wlbrb oetl nrb feksmhbrbh ns

vbfters mkCd). Ketb tlnt dntrmx duitmpimfntmek ms `bkbrniiy ket feddutntmvb,m.b., NO "ON. Mkhbbh, tlb prehuft NO dny ket bvbk ob hbcmkbh.

Bxndpib 6.>: Ibt N nkh O ob `mvbk oy

N 4

> 3 !?

6 8 2

0 :

> !6

"

#

$$$

%

&

'''4

>( ? + 3 ( 3 ! ? (> !>( = + 3 (>+ ? ( 66 ( ? + 8 ( 3 + 2 (> !6 ( = + 8 (>! 2 ( 6

0 ( ? + : ( 3 + < (> !0 ( = + : (>! < ( 6

"

#

$$$

%

&

'''

4

63

6

62 !6 :

!

"#

$

%&

tlbk

NO 4> ?

6 8

!

"#

$

%&

: >

> :

!

"#

$

%& 4

? >

8 6

!

"#

$

%&

wlmibON 4

: >

> :

!

"#

$

%&

> ?

6 8

!

"#

$

%& 4

6 8

> ?

!

"#

$

%&'NO . #

Bxndpib 6.>> Xwe etlbr spbfmni fnsbs ec dntrmx duitmpimfntmek nrb wertl bx-

pimfmtiydbktmekmk`. Ibt \ !Ckob tlb feiudk vbfter


36/55


\ 4

x>

!

xk

!

"

###

$

%

&&& .

Mc N ms nk d x k dntrmx, wb dny feksmhbr \ te ob nk k x > dntrmx nkh cerd tlbprehuft N\7

N\ 4

n>>

! n>k

" "

nd>

! ndk

!

"

###

$

%

&&&

x>

"

xk

!

"

###

$

%

&&&4

n>>x

>+!+n

>kxk

"

nd>x

>+

!

+ndkxk

!

"

###

$

%

&&&4

N> \

"

Nd \

!

"

###

$

%

&&& .

Ns bxpbftbh, tlb prehuft N\ ms nk d x > dntrmx wmtl bktrmbs `mvbk oy tlb stnk-

hnrhsfninr prehuft N\ mk Ckec tlb mtl rew ec N wmtl tlb vbfter \. Ketbtlnt tlms dny nise ob wrmttbk mk tlb cerd

N\ 4

n>>

!

nd>

!

"

###

$

%

&&&x>+"+

n>k

!

ndk

!

"

###

$

%

&&&xk

wlmfl fibnriy slews tlnt N\ ms gust n imkbnr fedomkntmek ec tlb feiudks ec N.

Kew ibt ] !Cdob tlb rew vbfter ] 4 (y, . . . , yd). Mc wb vmbw tlms ns n> x d dntrmx, tlbk wb dny cerd tlb > x k dntrmx prehuft ]N `mvbk oy

]N4 (y>,,yd )

n>> ! n>k

" "

nd> ! ndk

!

"

###

$

%

&&&

4 (y>n>> +!+ydnd>,,y>n>k +!+ydndk)

4 (] N>,,] Nk ) .

Xlms n`nmk ymbihs tlb bxpbftbh cerd ec tlb prehuft wmtl bktrmbs ]Nm. #

Xlms bxndpib su``bsts tlb ceiiewmk` feddekiy usbh ketntmek cer systbds

ec imkbnr bquntmeks. Feksmhbr tlb systbd

nmgxg 4ymg4>

k

!

wlbrb N 4 (n) ms nk d x k dntrmx. Vuppesb tlnt wb hbcmkb tlb feiudk vbfters


37/55


38/55


@mvbk n dntrmx N 4 (n), wb hbcmkb tlb trnkspesbec N, hbketbh oy NX4

(nXmg) te ob tlb dntrmx wmtl bktrmbs `mvbk oy nX 4 n. Mk etlbr werhs, mc N ms nk

d x k dntrmx, tlbk NXms nk k x d dntrmx wlesb feiudks nrb gust tlb rews ecN. Ketb mk pnrtmfuinr tlnt n feiudk vbfter ms gust tlb trnkspesb ec n rew vbfter.

Bxndpib 6.>? Mc N ms `mvbk oy> ? 6

8 2 3

!

"#

$

%&

tlbk NXms `mvbk oy> 8

? 2

6 3

!

"

###

$

%

&&& . #

Xlberbd 6.>


39/55

6.3 DNX_M\ NI@BO_N >26

(NO)m 4 (!

anmaoa>,,!anmaoar)

4 (nm>,,nmk )

o>> ! o>r

" "

ok> ! okr

"

#

$$$

%

&

'''4 N

mO .

Vmdminriy, cer tlb feiudks ec n prehuft wb cmkh tlnt tlb g tlfeiudk ec NO ms nimkbnr fedomkntmek ec tlb feiudks ec N7

(NO)g 4

!an>aoag!

!andaoag

"

#

$$$

%

&

'''4

n>a

!

nda

"

#

$$$

%

&

'''oag 4

a4>

k

( Naoaga4>

k

(

nkh

(NO)g 4

!an>aoag

!

!andaoag

"

#

$$$

%

&

'''4

n>> " n>k

! !

nd> " ndk

"

#

$$$

%

&

'''

o>g

!

okg

"

#

$$$

%

&

'''4NO g

.

Xlbsb cerduins wmii ob qumtb usbcui te us mk sbvbrni ec tlb ceiiewmk` tlberbds.

Xlberbd 6.>= Cer nky dntrmx N wb lnvb r(NX) 4 r(N).

Zreec Xlms ms Bxbrfmsb 6.3.?.

Xlberbd 6.?: Mc N nkh O nrb nky dntrmfbs cer wlmfl tlb prehuft NO mshbcmkbh, tlbk tlb rew spnfb ec NO ms n suospnfb ec tlb rew spnfb ec O, nkh tlb

feiudk spnfb ec NO ms n suospnfb ec tlb feiudk spnfb ec N.

Zreec Rsmk` (NO) 4 nO, wb sbb tlnt tlb mtlrew ec NO ms mk tlb spnfbspnkkbh oy tlb rews ec O, nkh lbkfb tlb rew spnfb ec NO ms n suospnfb ec tlbrew spnfb ec O.

Kew ketb tlnt tlb feiudk spnfb ec NO ms gust tlb rew spnfb ec (NO) X4

OXNX, wlmfl ms n suospnfb ec tlb rew spnfb ec NX oy tlb cmrst pnrt ec tlb

tlberbd. Out tlb rew spnfb ec NXms gust tlb feiudk spnfb ec N.

Fereiinry r(NO) dmk{r(N), r(O)}.

Zreec Ibt SNob tlb rew spnfb ec N, nkh ibt PNob tlb feiudk spnfb ec N.Xlbk

r(NO) 4 hmd SNO hmd SO 4 r(O)


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wlmibr(NO) 4 hmd PNO hmd PN 4 r(N) .

Bxbrfmsbs

>. Fedpibtb tlb preec ec Xlberbd 6.>0.

?. Zrevb Xlberbd 6.>=.

6. Ibt N ob nky d x k dntrmx nkh ibt \ ob nky k x > dntrmx, oetl wmtl

bktrmbs mkC. Hbcmkb tlb dnppmk` c 7CkCdoy c(\) 4 N\.(n) Vlew tlnt c ms n imkbnr trnkscerdntmek (m.b., n vbfter spnfb lededer-plmsd).

(o) Hbcmkb Md c 4 {N\7 \!Ck}. Vlew tlnt Md c ms n suospnfb ecCd.(f) Ibt R ob tlb feiudk spnfb ec N. Vlew tlnt Md c 4 R. QLmkt7 RsbBxndpib 6.>> te slew tlnt Md c R. Kbxt usb tlb bquntmek (NM)g4 NMgteslew tlnt R Md c.T(h) Ibt K hbketb tlb seiutmek spnfb te tlb systbd N\ 4 :. Mk etlbr

werhs, K 4 {\ !Ck7 N\ 4 :}. (K ms usuniiy fniibh tlb kuii spnfbec N.)Vlew tlnt

hmd K + hmd R 4 k .

QLmkt7 Vuppesb hmd K 4 r, nkh bxtbkh n onsms {x, . . . , x r} cer K te n

onsms {x} cer Ck. Vlew tlnt R ms spnkkbh oy tlb vbfters Nxr+> , . . . ,Nxk, nkh tlbk tlnt tlbsb vbfters nrb imkbnriy mkhbpbkhbkt. Ketb tlnt tlms

bxbrfmsb ms rbniiy gust nketlbr preec ec Xlberbd 6.>6.T

8. N dntrmx ec tlb cerd

n>>

: : ! :

: n??

: ! :

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: : : ! nkk

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ms fniibh n hmnèknidntrmx. Mk etlbr werhs, N 4 (n) ms hmnèkni mc n 4 :cer m "g. Mc N nkh O nrb oetl squnrb dntrmfbs, wb dny hbcmkb tlb feddu-tnterQN, OT ec N nkh O te ob tlb dntrmx QN, OT 4 NO - ON. Mc QN, OT 4 :,wb sny tlnt N nkh O feddutb.(n) Vlew tlnt nky hmnèkni dntrmfbs N nkh O feddutb.


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(o) Zrevb tlnt tlb ekiy k x k dntrmfbs wlmfl feddutb wmtl bvbry k x khmnèkni dntrmx nrb hmnèkni dntrmfbs.

2.

@mvbk tlb dntrmfbs3.

N 4

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fedputb tlb ceiiewmk`7(n) NO.(o) ON.

(f) NNX.

(h) NXN.

(b) Sbrmcy tlnt (NO)X4 OXNX.

3. Feksmhbr tlb dntrmx N !Dk(C) `mvbk oy

N 4

: > : : ! : :

: : > : ! : :

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.

Xlus N lns zbre bktrmbs bvbrywlbrb bxfbpt ek tlb supbrhmnèkniwlbrb

tlb bktrmbs nrb >s. Ibt N?4 NN, N64 NNN, nkh se ek. Vlew tlnt Nk4 :

out Nk-> ":.

0. @mvbk n dntrmx N 4 (n) !Dk(C), tlb sud ec tlb hmnèkni bibdbkts ec Nms fniibh tlb trnfbec N, nkh ms hbketbh oy Xr N. Xlus

XrN 4 nmm .

m4>

k

!

(n) Zrevb tlnt Xr(N + O) 4 Xr N + Xr O nkh tlnt Xr(N) 4 (Xr N) cernky sfninr .(o) Zrevb tlnt Xr(NO) 4 Xr(ON).

.


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43/55


>>. N squnrb dntrmx V ms snmh te ob syddbtrmfmc VX4 V, nkh n squnrb dntrmx

N ms snmh te ob sabwsyddbtrmf (er nktmsyddbtrmf) mc NX 4 -N. (Pbfektmkub te nssudb ns usuni tlntCms ket ec flnrnftbrmstmf?.)(n) Vlew tlnt V ": nkh N nrb imkbnriy mkhbpbkhbkt mk Dk(C).

(o) Plnt ms tlb hmdbksmek ec tlb spnfb ec nii k x k syddbtrmf dntrmfbs5(f) Plnt ms tlb hmdbksmek ec tlb spnfb ec nii k x k nktmsyddbtrmfdntrmfbs5

>?. Cmkh n onsms {N} cer tlb spnfb Dk(C) tlnt feksmsts ekiy ec dntrmfbs wmtl

tlb prepbrty tlnt N? 4 N (sufl dntrmfbs nrb fniibh mhbdpetbkt erpregbftmeks). QLmkt7 Xlb dntrmfbs

> :

: :

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wmii wera mk tlb pnrtmfuinr fnsb ec D?(C).T

>6. Vlew tlnt mt ms mdpessmoib te cmkh n onsms cer Dk(C) sufl tlnt bvbry pnmrec dntrmfbs mk tlb onsms feddutb wmtl bnfl etlbr.

>8. (n) Vlew tlnt tlb sbt ec nii keksmkùinr k x k dntrmfbs cerds n spnkkmk`sbt cer Dk(C). Bxlmomt n onsms ec sufl dntrmfbs.(o) _bpbnt pnrt (n) wmtl tlb sbt ec nii smkùinr dntrmfbs.

>2. Vlew tlnt tlb sbt ec nii dntrmfbs ec tlb cerd NO - ON he ket spnkDk(C). QLmkt7 Rsb tlb trnfb.T

>3. Ms mt pessmoib te spnk Dk(C) usmk` pewbrs ec n smkìb dntrmx N5 Mk etlbr

werhs, fnk {M , N, N?, . . . , Nk, . . .} spnk Dk(C)5 QLmkt7 FeksmhbrBxbrfmsb 8 noevb.T

6.0 MKSB_XMOIB DNX_MFBV

Pb sny tlnt n dntrmx N !D(C) ms keksmkùinrmc r(N) 4 k, nkh smkùinrmcr(N) 9 k. @mvbk n dntrmx N !D(C), mc tlbrb bxmsts n dntrmx O !D(C) sufltlnt NO 4 ON 4 M, tlbk O ms fniibh nk mkvbrsb ec N, nkh N ms snmh te obmkvbrtmoib.

Xbflkmfniiy, n dntrmx O ms fniibh n ibct mkvbrsb ec N mc ON 4 M, nkh ndntrmx O ms n rm`ltmkvbrsbec N mc NO 4 M. Xlbk, mc NO 4 ON 4 M, wb sny tlntO ms n twesmhbh mkvbrsb ec N, nkh N ms tlbk snmh te ob mkvbrtmoib.


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IMKBN_ B^RNXMEKV NKH DNX_MFBV>2 N dntrmx N !D(C) lns n rm`lt (ibct) mkvbrsb mc nkh ekiy mc N

ms keksmkùinr. Xlms rm`lt (ibct) mkvbrsb ms nise n ibct (rm`lt) mkvbrsb, nkh lbkfbms nk mkvbrsb ec N.

Zreec Vuppesb N lns n rm`lt mkvbrsb O. Xlbk NO 4 M se tlnt r(NO) 4 r(M).Vmkfb r(M) ms fibnriy bquni te k (Xlberbd 6.=), wb sbb tlnt r(NO) 4 k. Out tlbkcred tlb fereiinry te Xlberbd 6.?: nkh tlb cnft tlnt oetl N nkh O nrb k x kdntrmfbs (se tlnt r(N) k nkh r(O) k), mt ceiiews tlnt r(N) 4 r(O) 4 k, nkhlbkfb N ms keksmkùinr.

Kew suppesb tlnt N ms keksmkùinr se tlnt r(N) 4 k. Mc wb ibt Bgob tlb gtlfeiudk ec tlb mhbktmty dntrmx M, tlbk cer bnfl g 4 >, . . . , k tlb systbd ec

bquntmeks

Nmxm 4N\ 4Bg

m4>

k

!

lns n ukmqub seiutmek wlmfl wb hbketb oy \ 4 Og(Xlberbd 6.>3). Kew ibt O

ob tlb dntrmx wmtl feiudks Og. Xlbk tlb gtlfeiudk ec NO ms `mvbk oy

(NO)g 4 NOg 4 Bg

nkh lbkfb NO 4 M. Mt rbdnmks te ob slewk tlnt ON 4 M. Xe sbb tlms, ketb tlnt

r(NX) 4 r(N) 4 k (Xlberbd 6.>=) se tlnt NXms keksmkùinr nise. Lbkfb nppiy-mk`tlb sndb nrùdbkt slews tlbrb bxmsts n ukmqub k x k dntrmx F Xsufl tlnt

NX FX4 M. Vmkfb (FN)X4 NXFXnkh MX4 M, tlms ms tlb sndb ns FN 4 M.Pb kew rbfnii tlnt mt wns slewk prmer te tlb tlberbd tlnt mc N lns oetl n ibctnkh n rm`lt mkvbrsb, tlbk tlby nrb tlb sndb. Xlbrbcerb O 4 F se tlnt ON 4 NO4 M, nkh lbkfb O ms nk mkvbrsb ec N. Fibnriy, tlb preec rbdnmks vnimh mc rm`ltms rbpinfbh oy ibct tlreu`leut.

Fereiinry > N dntrmx N ! Dk(C) ms keksmkùinr mc nkh ekiy mc mt lns nkmkvbrsb. Curtlbrderb, tlms mkvbrsb ms ukmqub.

Zreec Ns wb snw noevb, mc O nkh F nrb oetl mkvbrsbs ec N, tlbk O 4 OM 4O(NF) 4 (ON)F 4 MF 4 F.

Mk vmbw ec tlms fereiinry, tlb ukmqub mkvbrsb te n dntrmx N wmii ob hbketbhoy N cred kew ek.


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6.0 MKSB_XMOIB DNX_MFBV >2=

Fereiinry ? Mc N ms nk k x k keksmkùinr dntrmx, tlbk N ms keksmkùinr nkh(N) 4 N.

Zreec Mc N ms keksmkùinr, tlbk (oy Xlberbd 6.?>) N bxmsts se tlnt NN 4NN 4 M. Out tlms dbnks tlnt (N) bxmsts nkh ms bquni te N, nkh lbkfb N ms

nise keksmkùinr.

Fereiinry 6 Mc N nkh O nrb keksmkùinr tlbk se ms NO, nkh (NO) 4 O N.

Zreec Xlb cnft tlnt N nkh O nrb keksmkùinr dbnks tlnt N nkh O bxmst.Pb tlbrbcerb sbb tlnt

(ON)(NO) 4 OMO 4 OO 4 M

nkh smdminriy (NO)(ON) 4 M. Mt tlbk ceiiews tlnt O N 4 (NO). Vmkfb wb

lnvb kew slewk tlnt NO lns nk mkvbrsb, Xlberbd 6.?> tbiis us tlnt NO dustob keksmkùinr.

Fereiinry 8 Mc N ms keksmkùinr tlbk se ms NX, nkh (NX) 4 (N)X.

Zreec Xlnt NXms keksmkùinr ms n hmrbft feksbqubkfb ec Xlberbd 6.>=. Kbxtwb eosbrvb tlnt

(N)XNX 4 (NN)X 4 MX 4 M

se tlnt tlb ukmqubkbss ec tlb mkvbrsb tbiis us tlnt (NX) 4 (N)X. Ketb tlms

nise slews tlnt NXms keksmkùinr.

Fereiinry 2 N systbd ec k imkbnr bquntmeks mk k ukakewks lns n ukmqubseiutmek mc nkh ekiy mc mts dntrmx ec febccmfmbkts ms keksmkùinr.

Zreec Feksmhbr tlb systbd N\ 4 ]. Mc N ms keksmkùinr, tlbk n ukmqub Nbxmsts, nkh tlbrbcerb wb lnvb \ 4 N] ns tlb ukmqub seiutmek. (Ketb tlnt tlmsms bssbktmniiy tlb fektbkt ec Xlberbd 6.>3.)

Fekvbrsbiy, mc tlms systbd lns n ukmqub seiutmek, tlbk tlb seiutmek spnfb ectlb nssefmntbh lede`bkbeus systbd dust lnvb hmdbksmek : (Xlberbd 6.>2).Xlbk Xlberbd 6.>6 slews tlnt wb dust lnvb r(N) 4 k, nkh lbkfb N ms kek-smkùinr.

N dnger preoibd tlnt wb lnvb ket ybt hmsfussbh ms lew te nftuniiy cmkh tlbmkvbrsb ec n dntrmx. Ekb dbtleh mkveivbs tlb usb ec hbtbrdmknkts ns wb wmiisbb mk tlb kbxt flnptbr. Lewbvbr, ibt us slew nketlbr npprenfl onsbh ek tlbcnft tlnt n keksmkùinr dntrmx ms rew-bqumvnibkt te tlb mhbktmty dntrmx


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(Xlberbd 6.>:). Xlms dbtleh lns tlb nhvnktn`b tlnt mt ms ni`ermtldmf, nkhlbkfb ms bnsmiy mdpibdbktbh ek n fedputbr.

Vmkfb tlb gtlfeiudk ec n prehuft NO ms NOg, wb sbb tlnt feksmhbrmk` tlbpnrtmfuinr fnsb ec NN 4 M ibnhs te

(NN)g 4 N(N)g 4 Bg

wlbrb Bg ms tlb gtlfeiudk ec M. Plnt wb kew lnvb ms tlb keklede`bkbeussystbd

N\ 4 ]

(er nx 4 y) wlbrb \ 4 (N)gnkh ] 4 Bg. Ns wb snw mk Vbftmek 6.?, wbdny seivb cer tlb vbfter \ oy rbhufmk` tlb nu`dbktbh dntrmx te rbhufbh rew-bflbiekcerd. Cer tlb pnrtmfuinr fnsb ec g 4 > wb lnvb

nu` N 4

n>> ! n>k >

n?> ! n?k :

" " "

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nkh lbkfb tlb rbhufbh cerd wmii ob

> : : ! : f>>

: > : ! : f?>

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: : : ! > fk>

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cer sedb sbt ec sfninrs f. Xlms dbnks tlnt tlb seiutmek te tlb systbd ms x 4

f, x 4 f, . . . , x 4 f. Out \ 4 (N)>4 tlb cmrst feiudk ec N, nkhtlbrbcerb tlms inst dntrmx dny ob wrmttbk ns

> ! : n!>>>

" " "

: ! > n!>k>

"

#

$

$$

%

&

'

'' .

Kew, cer bnfl g 4 >, . . . , k tlb systbd N\ 4 N(N)g4 Bgniwnys lns tlbsndb dntrmx ec febccmfmbkts, nkh ekiy tlb inst feiudk ec tlb nu`dbktbh dntrmxhbpbkhs ek g. Vmkfb cmkhmk` tlb rbhufbh rew-bflbiek cerd ec tlb dntrmx ec


47/55


48/55


Bxbrfmsbs

>. Sbrmcy tlb rbhufbh rew-bflbiek cerd ec tlb dntrmx `mvbk mk Bxndpib6.>6.

?. Cmkh tlb mkvbrsb ec n `bkbrni ? x ? dntrmx. Plnt fekstrnmkts nrb tlbrb ektlb bktrmbs ec tlb dntrmx5

6. Vlew tlnt n dntrmx ms ket mkvbrtmoib mc mt lns nky zbre rew er feiudk.

8. Cmkh tlb mkvbrsb ec bnfl ec tlb ceiiewmk` dntrmfbs7

(n)

> : ?

? !> 6

8 > :

?x + 2y+ ?z 4>8

x + 6y + 6z 46:

(o) x + ?y+ z 4 ?

?x + 2y+ ?z 4 !>

x + 6y+ 6z 4 3

3. Plnt ms tlb mkvbrsb ec n hmnèkni dntrmx5

0. (n) Zrevb tlnt nk uppbr-trmnkùinrdntrmx ms mkvbrtmoib mc nkh ekiy mc bvbrybktry ek tlb dnmk hmnèkni ms kekzbre (sbb Bxbrfmsb 6.3.= cer tlb hbcmkm-tmekec nk uppbr-trmnkùinrdntrmx).(o) Zrevb tlnt tlb mkvbrsb ec n iewbr (uppbr) trmnkùinr dntrmx ms iewbr(uppbr) trmnkùinr.

? 6 8

: ? 6 8

: : 6 8

: : : 8

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&&&&

.

=. (n) Ibt N ob nky ? x > dntrmx, nkh ibt O ob nky > x ? dntrmx. Zrevb tlntNO ms ket mkvbrtmoib.


49/55


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51/55

6.< BIBDBKXN_] DNX_MFBV >32

te rew m. Xlus nii tlrbb typbs ec bibdbktnry rew epbrntmeks lnvb mkvbrsbswlmfl nrb nise bibdbktnry rew epbrntmeks.

Oy wny ec kedbkfinturb, n squnrb dntrmx N 4 (n) ms snmh te ob hmnèknimcn 4 : cer m "g. Xlb dest feddek bxndpib ec n hmnèkni dntrmx ms tlb mhbktmtydntrmx.

Xlberbd 6.?6 Bvbry bibdbktnry dntrmx ms keksmkùinr, nkh

Qb(M)T 4 b(M) .

Curtlbrderb, tlb trnkspesb ec nk bibdbktnry dntrmx ms nk bibdbktnry dntrmx.

Zreec Oy hbcmkmtmek, b(M) ms rew bqumvnibkt te M nkh lbkfb lns tlb sndb rnkans M (Xlberbd 6.8). Xlus b(M) ms keksmkùinr smkfb r(M) 4 k, nkh lbkfb b(M)bxmsts. Vmkfb mt wns slewk noevb tlnt b ms nk bibdbktnry rew epbrntmek, wb

nppiy Xlberbd 6.?? te tlb dntrmx b(M) te eotnmk

b(M)b(M) 4 b(b(M)) 4 M .

Vmdminriy, nppiymk` Xlberbd 6.?? te b(M) ymbihs

b(M)b(M) 4 b(b(M)) 4 M .

Xlms slews tlnt b(M) 4 Qb(M)T.Kew ibt b ob n typb trnkscerdntmek tlnt mktbrflnk`bs rews m nkh g (wmtl

m 9 g). Xlbk tlb mtlrew ec b(M) lns n > mk tlb gtlfeiudk, nkh tlb gtlrew lns n> mk tlb mtlfeiudk. Mk etlbr werhs,

Qb(M)T 4 > 4 Qb(M)T

wlmib cer r, s "m, g wb lnvb

Qb(M)Trs4 : mc r "s

nkh

Qb(M)Trr 4 > .

Xnamk` tlb trnkspesb slews tlnt

Qb(M)TX 4 Qb(M)T 4 > 4 Qb(M)Tnkh

Qb(M)TXrs 4 Qb(M)Tsr 4 : 4 Qb(M)Trs .


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53/55

6.< BIBDBKXN_] DNX_MFBV >30

ec tlb tlrbb typbs ec bibdbktnry rew epbrntmeks, nkh ibt N ob nk k x k dntrmx.Ns wb lnvb sbbk, duitmpiymk` N cred tlb ibct oy B rbsuits mk n kbw dntrmxwmtl tlb sndb rews tlnt weuih rbsuit cred nppiymk` tlb bibdbktnry rew

epbrntmek te N hmrbftiy. Pb finmd tlnt duitmpiymk` N cred tlb rm`lt oy BXrbsuits mk n kbw dntrmx wlesb feiudks lnvb tlb sndb rbintmekslmp ns tlb rewsec BN. Pb wmii prevb tlms cer n typb epbrntmek, ibnvmk` tlb bnsmbr typb nkh$epbrntmekste tlb rbnhbr (sbb Bxbrfmsb 6.).

Ibt ob tlb nhhmtmek ec f tmdbs rew g te rew m. Xlbk tlb rews ec B nrb

`mvbk oy B 4 M cer a "m, nkh B 4 M + fM. Xlbrbcerb tlb feiudks ec BXnrb`mvbk oy

(BX)a 4 Ma cer a "mnkh

(BX)m 4 Mm+ fMg .

Kew rbfnii tlnt tlb atlfeiudk ec NO ms `mvbk oy (NO)a4 NO

a. Pb tlbk lnvb

(NBX)a 4 N(BX)a 4 NMa 4 Na cer a "mnkh

(NBX)m 4 N(BX)m 4 N(Mm+ fMg) 4 NMm+ fNMg 4 Nm+ fNg .

Xlms ms tlb sndb rbintmekslmp ns tlnt ceukh obtwbbk tlb rews ec BN wlbrb(BN) 4 N nkh (BN) 4 N + fN (sbb tlb preec ec Xlberbd 6.??).

Bxbrfmsbs

>. Ibt N ob nk k x k dntrmx, nkh ibt B ob nk k x k bibdbktnry dntrmx rbprb-

sbktmk`n typb er $epbrntmek. Vlew tlnt tlb feiudks ec NBXlnvb tlbsndb rbintmekslmp ns tlb rews ec BN.

?. Prmtb hewk 8 x 8 bibdbktnry dntrmfbs tlnt wmii mkhufb tlb ceiiewmk` bib-dbktnryepbrntmeks mk n 8 x 8 dntrmx wlbk usbh ns ibctduitmpimbrs. Sbrmcytlnt yeur nkswbrs nrb ferrbft.(n) Mktbrflnk`b tlb ?khnkh 8tlrews ec N.

(o) Mktbrflnk`b tlb ?khnkh 6rhrews ec N.(f) Duitmpiy tlb 8tlrew ec N oy 2.(h) Nhh a tmdbs tlb 8tlrew ec N te tlb >strew ec N.(b) Nhh a tmdbs tlb >strew ec N te tlb 8tlrew ec N.

6. Vlew tlnt nky b(N) dny ob wrmttbk ns n prehuft ec b!(N)s nkh b(N)s.(Xlb ketntmek sleuih ob eovmeus.)


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:

?

: 6 !>

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'''

te tlb rbhufbh rew-bflbiekcerd _, nkh wrmtb tlb bibdbktnry dntrmx fer-rbspekhmk` te bnfl ec tlb bibdbktnry rew epbrntmeks rbqumrbh. Cmkh n

keksmk`uinr dntrmx Z sufl tlnt ZN 4 _ oy tnamk` tlb prehuft ec tlbsb bib-dbktnrydntrmfbs.

0. Ibt N ob nk k x k dntrmx. Vuddnrmzb sbvbrni ec eur rbsuits oy prevmk`tlnt tlb ceiiewmk` nrb bqumvnibkt7(n) N ms mkvbrtmoib.(o) N ms rew bqumvnibkt te M .(f) N ms n prehuft ec bibdbktnry dntrmfbs.

:. Vlew tlnt nk d x k rbhufbh rew-bflbiek dntrmx _ ec rnka a fnk obrbhufbh oy bibdbktnry feiudk epbrntmeks te nk d x k dntrmx F ec tlbcerd


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6.< BIBDBKXN_] DNX_MFBV >3=

F 4

> : ! : ! : :

: > ! : ! : :

" " " " "

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&&&&

wlbrb tlb cmrst a bktrmbs ek tlb dnmk hmnèkni nrb >s, nkh tlb rbst nrb :s.

>>. Cred tlb prbvmeus preoibd nkh Xlberbd 6.6, slew tlnt bvbry d x kdntrmx N ec rnka a fnk ob rbhufbh oy bibdbktnry rew nkh feiudk epbrn-tmekste tlb cerd F. Pb fnii tlb dntrmx F tlb fnkekmfni cerdec N.

>?. Pb sny tlnt n dntrmx N ms rewfeiudkbqumvnibkt(noorbvmntbh oy r.f.b.)te n dntrmx O mc N fnk ob trnkscerdbh mkte O oy n cmkmtb kudobr ec bib-dbktnryrew nkh feiudk epbrntmeks. Zrevb7(n) Mc N ms n dntrmx, b ms nk bibdbktnry rew epbrntmek, nkh b ms nkbibdbktnryfeiudk epbrntmek, tlbk (bN)b 4 b(Nb).(o) r.f.b. ms nk bqumvnibkfb rbintmek.(f) Xwe d x k dntrmfbs N nkh O nrb r.f.b. mc nkh ekiy mc tlby lnvb tlbsndb fnkekmfni cerd, nkh lbkfb mc nkh ekiy mc tlby lnvb tlb sndb rnka.

>6. Mc N ms nky d x k dntrmx ec rnka a, prevb tlnt tlbrb bxmsts n keksmkùinrd x d dntrmx Z nkh n keksmkùinr k x k dntrmx ^ sufl tlnt ZN^ 4 F (tlbfnkekmfni cerd ec N).

>8. Zrevb tlnt twe d x k dntrmfbs N nkh O nrb r.f.b. mc nkh ekiy mc tlbrb bxmstsn keksmkùinr d x d dntrmx Z nkh n keksmkùinr k x k dntrmx ^ sufl tlntZN^ 4 O.

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Lineer Cebir - Linear Equations and Matrices