matematicas espacios vectoriales (by carrascal)

11
Aôdha Mdrrdsmdk C`z`. Ispdma`s vimt`radkis. _r`nkicds ji Q dl``! ^ispuistds v4.: "#"#4$._%&a'd 4 ji 44 Ispdma`s vimt`radkis. Aôdha Mdrrdsmdk C`z`.   gãsamdsOydl``.is  lttp6//www .mdrrdsmdk.bit:7.bit/gã samds/ v4.?6 >>/44/>?4>. Öktacd dmtudkazdmaýb6 v4.:6 ?/?>/>?4! (ut`r Aôdha Mdrrdsmdk C`z` )*tuk` jik j`muci't` ji Cdtic"tamds Ispdma`s vimt`radkis +i ml d mri dm a, ' - / pri &u' td s r is ui kt ds " p%& a' ds mu d' j` si mr i, ? 4 . - ##" 44" # 4# 4 4 pri &u' td s " > p% &. +i ml d d mt ud ka zd ma ,' - / p ri &u 't ds risu ik td s " p% &a 'd s d mt ud ka jd j ? 4. : - " #" # 4$ 4= pri& u'td s " 44 p %& . I'kdmi 9a'h7 jik j`muci't` lttp5""is.smranj.m`c"A<M$<14dhaMdrrdsmdk lttp5""is.smranj.m`c"j`m"4$68=:$4"Cdticdtamds-Ispdma`s-?imt`radkis-n;-Mdrrdsmdk  _uist` =¶ ib K`s cif`ris ji Gãsamd–. 1= % Cif`r ^ispuistd. <? gdbs. a!i" #. $. &<? pubt`s. '#1 pri(ubtds )?#*?$*$?'=+   , 3 ( d - m j )   #  / tr (  , )   + jit ,3  , 3 d - m j i g   ( ? a ∞θ $ + (tr , ) θ #  / X d i+ d a+ i a / ( - j +  m ( +  g ?)S θ + jit , 3 Aôdha Mdrrdsmdk C`z` v4.5 ##"44"#4# lttp5""333.mdrrdsmdk. 'it:=.'it"2*samds" 2*samds0;dl``.is  Aôdha Mdrrdsmdk

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7/23/2019 Matematicas Espacios Vectoriales (by Carrascal)

http://slidepdf.com/reader/full/matematicas-espacios-vectoriales-by-carrascal 1/11

Aôdha Mdrrdsmdk C`z`. Ispdma`s vimt`radkis. _r`nkicds ji Qdl``! ̂ ispuistds v4.: "#"#4$._%&a'd 4 ji 44

Ispdma`s vimt`radkis. Aôdha Mdrrdsmdk C`z`.  gãsamdsOydl``.is lttp6//www.mdrrdsmdk.bit:7.bit/gãsamds/ 

v4.?6 >>/44/>?4>. Öktacd dmtudkazdmaýb6 v4.:6 ?/?>/>?4!

(ut`r Aôdha Mdrrdsmdk C`z`

)*tuk` jik j`muci't` ji Cdtic"tamds Ispdma`s vimt`radkis

+imld mridma,' - / pri&u'tds risuiktds " p%&a'ds mud'j` si mri, ? 4. - ##"44"#4# 44 pri&u'tds " > p%&.

+imld dmtudkazdma,' - / pri&u'tds risuiktds " p%&a'ds dmtudkajdj ? 4.: - "#"#4$ 4= pri&u'tds " 44 p%&.

I'kdmi 9a'h7 jik j`muci't` lttp5""is.smranj.m`c"A<M$<14dhaMdrrdsmdk

lttp5""is.smranj.m`c"j`m"4$68=:$4"Cdticdtamds-Ispdma`s-?imt`radkis-n;-Mdrrdsmdk

 _uist` =¶ ib ‛K`s cif`ris ji Gãsamd–. 1= % Cif`r ^ispuistd. <? gdbs. a!i" #. $.&<? pubt`s. '#1 pri(ubtds )?#*?$*$?'=+

   , 3 (d -

m j )

 # / tr ( ,) + jit ,3

 , 3d - m

j i g  

 ( ? a

∞θ$+ (tr ,) θ# / X di+ d a+ i a / (- j + m ( +   g ?)Sθ+ jit ,3

Aôdha Mdrrdsmdk C`z` v4.5 ##"44"#4# lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  Aôdha Mdrrdsmdk

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Aôdha Mdrrdsmdk C`z`. Ispdma`s vimt`radkis. _r`nkicds ji Qdl``! ̂ ispuistds v4.: "#"#4$._%&a'd # ji 44

  Ik 44 ji `mtunri ji #4# ci lami caicnr` ji Qdl``! ^ispuistds ; ci pusi d ris`kvir

iOirmama`s Dui pkd'tidnd kd m`cu'ajdj. Ci &ustdnd ik pkd'tidcai't` ji Dui dk&u'd jujd Dui tuvaird

dk&uai' pujaird ris`kvirkd u'` ` vdra`s m`cpdôir`s d trdvNs ji a'tir'it. I' idmtdci'ti7 u' cis

dkmd'mN ik avik = supird'j` k`s 4. pu't̀ s7. Q i' Oustdci'ti7 j`s cisis li supirdj` k`s

4. pu't`s ; ci li m`kdj` ldstd ik !# puist` $i k`s %Ci&`ris $i '(samd). Jisji ik ji2inrir` ji #4$ `mup` ik *avik . `; GM`kd+`rd$`r $istdmd$`E i' '(samdL Cdtic"tamds  i

Ab,ibair(d ; tdcnaN' k` 2ua i' -`tdri7. Jisji ik #$ ji '`vaicnri ist`; i'tri k`s 4 pracir`s

 puist`s ji G9`s ciO`ris ji +*samdEL ik $ ji jamaicnri `mupN ik / puist`L ik 46 ji jamaicnri ik 6/

ku&dr ; ik ji 2inrir` ji #4$L ik tirmir puist`. i dkmd'zdj` u' !  ji Ci&`r ispuistd.

( 2imld "#"#4$ kkiv` m`'tistdjds 468 pri&u'tdsL m`' #6.=> pu't`s 'avik 7L ti'&` u'

8$ < ji CiO`r ^ispuistdL `mup` kd $F p`sama,' ji G9`s CiO`ris ji +*samdE ; ti'&` =? %dbs).

( m`'ta'udma,' prisi't` u'ds mud'tds pri&u'tds ji Qdl``! ^ispuistds Dui ;` casc`

ris`kv*L m`' kd 2imld i' Dui k` lami. I' kd cd;`r*d ji ikkds li rispitdj` kd pri&u'td `ra&a'dk. Huaz%

ld;d m`rri&aj` dk&u'd 2dktd ji `rt`&rd2*d ` ciO`rdj` u' p`m` ik i'u'madj` pdrd ldmirk` cds mkdr`.

M`' ik taicp` arN `rji'd'j` k`s iOirmama`s p`r `rji' ji ja2amuktdj mrimai'ti.

(u'Dui cuml`s pr`nkicds puiji' ris`kvirsi ji `trd 2`rcdL mri` Dui kds rispuistds Dui li

jdj` s`' kd cd'ird c%s djimudjd ji ldmirk L̀ 2rut` ji dô`s ji j`mi'mad ji +*samd ; Cdtic%tamds d

'avik u'avirsatdra`.

9ds rispuistds ist%' tdk ; m`c` kds risp`'j* i' su c`ci't`L m`' ik si'makk`7 ijat`r ji

tit`s ji Qdl``. i pr`murdj` p`'ir pdrN'tisas pdrd mkdra2amdr kds cascds. 9ds p`ti'mads kds a'jam`

m`'5 K 6K:7 kd rd*z mudjrdjdL m`' K4"#7L m`' H )̂ ` m`' GrdazE iO. rdaz #7 B #K4"#77 kd a'ti&rdk

m`c` Ga'ti&rdkE. ` puij` icpkidr sun*'jamis iO. rdja` ji kd )airrd5 ^) Npsak`' sun `7...

a dk&u'd ji kds rispuistds is a'm`rrimtdL Dui puijiL puis t`j`s s`c`s lucd'`s ; ji viz i'

mud'j` '`s m`'2u'jac`sL p`jNas m`cu'am%rcik` p`r m`rri` ikimtr,'am`5 2*samds0;dl``.is  .^ivasdrN ca rispuistd ; kd Dui ci a'jaDuNas ; kd m`rri&arN sa is 'imisdra`.

Ik 4>"44"#4# ci d&ri&uN dk `r`4??mad  ; ik 8"4#"#4# dk @rup` ji +dmin``h

9u$7c`sb`s ib cdtic"tamds p`r k` Dui tdcnaN' li a'mkuaj` cas rispuistds d iOirmama`s Dui li

risuikt` d trdvNs jik 2`r` ` ji +dmin``h.

)rds isti piDuiô` pr,k`&`L a'jam` kds 2,rcukds c%s usudkis Dui dpdrimi' i' ik ticd. M`' ik

taicp` Duaz% ipkaDui m,c` usdrkds ; m,c` ris`kvir jitirca'dj`s Gpr`nkicds c`jik`E. )dcnaN'

siôdk` `trds simma`'is rikdma`'djds m`' ik mdp*tuk̀ Dui istdc`s istujad'j` m`' su i'kdmi dk

j`muci't` ji mranj m`rrisp`'jai'ti. aicpri p`jNas rimurrar d ca p%&a'd 3in pdrd m`'suktdr cds

2,rcukds ; jdt`s5 lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  ; d ahapijad5 lttp5""is.3ahapijad.`r&" .

_`jNas ja2u'jar isti j`muci't` i'tri vuistr`s m`cpdôir`s. ^ispitdjL is` s*L ik 2`rcdt` jik

casc`L su dut`r*d ; kd cdrmd ji d&ud m`' ca 2`t`. T( Dui c`k` ji piDuiô`

Isti j`muci't` si istdr% dmtudkazd'j` m`'ta'udci'ti p`r k` Dui is rim`ci'jdnki Dui

m`'suktNas kd ]ktacd virsa,' dmtudkazdjd m`' 'uiv`s iOirmama`s7 jisji ca pir2ak i' mranj5

lttp5""is.smranj.m`c"A<M$<14dhaMdrrdsmdk

@rdmads d Duai'is ldnNas pri&u'tdj` ; risp`'jaj` tdcnaN'. Z' sdkuj` d t`j`s ; ispir` Dui

ist`s pr`nkicds `s puijd' sirvar i' vuistr`s istuja`s.

 4>da :drrdsmd" C`9`. 7isji :dstra""` ji 7`b 5udb ub piui>` pui-"` ji _d"ibmad )6spd>d+

8a uir;as !asatdr ca pui-"` imdj ubd `fidjd d ca 3i-2 ttp2**333.mdstra""`jij`bfudb.bitda.bit* 

Aôdha Mdrrdsmdk C`z` v4.5 ##"44"#4# lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  Aôdha Mdrrdsmdk

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Aôdha Mdrrdsmdk C`z`. Ispdma`s vimt`radkis. _r`nkicds ji Qdl``! ̂ ispuistds v4.: "#"#4$._%&a'd $ ji 44

5tr`s ticds rikdma`bd$`s m`b isti mdp(tuk`dmai'j` mkam s`nri ik i'kdmiL ti kkivd d ca j`muci't` ji mranj m`rrisp`'jai'ti

(ti'ma,'5 ik 4:"4"#4$ si javaji ik ticd ji ispdma s̀ vimt r̀adkisL dpkamdma`'is ka'idkis ;i'j`c`r2asc`s i' tris mdp*tuk`s a'jipi'jai'tis.

• 4. Ispdma`s vimt`radkis isti mdp*tuk`7

• #. (pkamdma`'is ka'idkis

• $. Jad&`'dkazdma,' ji i'j`c`r2asc`s

Cas rispuistds d kds pri,ubtds $i <d1`` ispuistdsv 4.: 8?/?>/>?4!. 4; pri,ubtds3

Ispdma`s vimt`radkis

 _ri(ubtd ?'. ^isui"td i" $=*'?*$?'$

-id ub ispdma` vimt`radk 2 9 ub m`b&ubt` 8u0 v0 3 kabidkcibti ab$ipib$aibti. Oicuistrd Duiik m`b&ubt` 8u N v0 v N 0 u N 3 is kabidkcibti ab$ipib$aibtiAôdha lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  lttp5""is.smranj.m`c"AôdhaMdrrdsmdk

^ispuistds 47

  Aôdha Aôdha

  Z' @i'a` is u' usudra` nrakkd'ti i' u'd mdti&`r*d.

  a u v 37 is ka'idkci'ti a'jipi'jai'tiL d u Y n v Y m 3 B sa ; s`k` sa dBnBmB

  ?idc`s sa u Y v v Y 3 u Y 37 is ka'idkci'ti a'jipi'jai'ti ` '`

  a k` is

  ( uYv7Y 1 vY37 Y M uY37 B sa ; s`k` sa (B1BMB

  ( u Y ( v Y 1 v Y 1 3 Y M u Y M 3 B

  (YM7 u Y (Y17 v Y 1YM7 3

  sa kkdcdc`s (YMBd L (Y1Bn ; 1YMBmL ti'ic`s Dui d u Y n v Y m 3 B ; lic`s jaml` d'tis Dui

dk sir ka'idkci'ti a'jipi'jai'tis dBnBmB

  ds* puis (Y M B L (Y1BL 1YMBL ji j`'ji sa ris`kvic`s (B1BMB

  Dui is k` Dui Duir*dc`s jic`strdr.

 _ri(ubtd ?$. ^isui"td i" ?0*''*$?'$

dkkdr ik vimt`r $i m``r$ibd$ds pdrd m`b rikdmaýb d kd +dsi -J 8u4 u>3d3 u4J 84 ?3 u> J 8? 43 J 8! G3+3 u> J 84 43 u> J 8? >3 J 8d +3

Aôdha lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  lttp5""is.smranj.m`c"AôdhaMdrrdsmdk

^ispuistds 47

  Aôdha Aôdha

  Z' @i'a` is u' usudra` nrakkd'ti i' u'd mdti&`r*d.

  d7 3 B W u4 Y ; W u#

  j`'ji L ;7 s`' kds m``rji'djds jik vimt`r 3 i' kd ndsi u4L u#

Aôdha Mdrrdsmdk C`z` v4.5 ##"44"#4# lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  Aôdha Mdrrdsmdk

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Aôdha Mdrrdsmdk C`z`. Ispdma`s vimt`radkis. _r`nkicds ji Qdl``! ̂ ispuistds v4.: "#"#4$._%&a'd : ji 44

  $L -7 B 4L 7 Y ; L 47L ji j`'jiL ̀ nvadci'tiL L ;7 B $L -7

  n7 Jik casc` c`j`5

  dL n7 B 4L 47 Y ; L #7 ; ̀ ntai'is L ; i' 2u'ma,' ji d ; n

  dL n7 B L Y # ;7

  ji j`'ji B d  Y # ; B n ; B n-d7 " #

  kui&` si trdtdr% jik vimt`r5 L ;7 B dL n-d7 " # 7

 _ri(ubtd ?=. ^isui"td i" '?*''*$?'$

EOitircabi sa isti m`b&ubt` riprisibtd ubd +dsi pdrd ispdma` vimt`radk LG> N N > > > N > N ! : > G 4F pdrd H>KAôdha lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  lttp5""is.smranj.m`c"AôdhaMdrrdsmdk

^ispuistds $7

  Aôdha Aôdha  Z' @i'a` is u' usudra` nrakkd'ti i' u'd mdti&`r*d.

  I' kd ndsi UK# L L 4\L k`s p`ka'`ca`s d'tira`ris k`s puijis ismranar m`c` vimt`ris

  i4 B -4L 4L #7L i# #L #L $7L i$ :L L -47

  k`s m`k`mds m`c` m`kuc'ds ` 2akds7 ji u'd cdtraz ; sa ik jitirca'd'ti is jasta't` ji mir` s`'

9AI(9CI)I AJI_IJAI)I ; +V^C( 1(I.

  I' i2imt` p`'&` kds m`cds pdrd sipdrdr k`s 'ucir` i'tri s*7

  !-4L 4L #!

  !#L #L $! B ( ca ci ld sdkaj` mir`L kui&` V +V^C( 1(I

  !:L L -4!

  ` 2`rcd' ndsi ji t`j` jik m`'Ou't` ji p`ka'`ca`s ji &rdj` ci'`r ` a&udk Dui j`s. +`rcdr*d'

u'd ndsi ji u' sunispdma` ji jaml` ispdma` vimt`radk ikaca'dc`s u'` ji k`s vimt`risp`ka'`ca`s7 ; k`s j`s ristd'tis sirad' ka'idkci'ti a'jipi'jai'tis ; mridr*d' u' sunispdma` '` t`j`

ik ispdma` vimt`radk.

 _ri(ubtd ?0. ^isui"td i" $<*''*$?'$

-a 2 J L 8936 9 J > N 4 pirtibimaibti d F is $imar 2 is ik m`b&ubt` $i pubt`s s`+ri kdrimtd 9 J > N 4. Is 2 ub ispdma` vimt`radk KAôdha lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  lttp5""is.smranj.m`c"AôdhaMdrrdsmdk

^ispuistds 47

  Aôdha Aôdha

  Z' @i'a` is u' usudra` nrakkd'ti i' u'd mdti&`r*d.

  `.

  _ruind p`r iOicpk` t`cd'j` j`s pu't`s ji kd rimtd

  sa B L ; B 4L ( L 47 pirti'imi dk sunispdma`

  sa B 4 L ; B $L 1 4L $7 pirti'imi dk sunispdma`

  vidc`s sa ( Y 1 pirti'imi dk sunispdma` ` '`

  ( Y 1 B 4L :7 vic`s Dui '` pirti'imi dk sunispdma` puis '` vira2amd su imudma,'

  : TB # W 4 Y 4

  vis Dui '`. 9ui&` '` is sunispdma` vimt`radk.

  _drd Dui k` 2uird si jini' vira2amdr j`s m`sds5

  A7 pdrd t`j` L ; pirti'imai'ti d ?L Y ; jini pirti'imir d ? Nstd lic`s vast` Dui '` kd vira2amd7

Aôdha Mdrrdsmdk C`z` v4.5 ##"44"#4# lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  Aôdha Mdrrdsmdk

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Aôdha Mdrrdsmdk C`z`. Ispdma`s vimt`radkis. _r`nkicds ji Qdl``! ̂ ispuistds v4.: "#"#4$._%&a'd 6 ji 44

  AA7 pdrd t`j` pirti'imai'ti d ?L pdrd t`j` kd'jd pirti'imai'ti dk muirp` j`'ji trdndOic`s

']cir`s ridkisL p`r iOicpk`7L kd'jd W jini pirti'imir dk sunispdma` ?.

 _ri(ubtd ?. ^isui"td i" '$*?'*$?'=. Ca rispuistd bOcir` '0&& 

dkkdr kd $acibsaýb 9 ubd +dsi $i k`s sa,uaibtis su+ispdma`s6v46ik m`b&ubt` $i cdtramis sac7tramds ib C>83v>6ik m`b&ubt` $i cdtramis dbtasac7tramds ib C>83

Aôdha lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  lttp5""is.smranj.m`c"AôdhaMdrrdsmdk

^ispuistds #7

  C`z` C`z`

  Z' @i'a` is u' usudra` nrakkd'ti i' u'd mdti&`r*d.

  u'd cdtraz sacNtramd (Kt B (7 ji ̀ rji' # is ji kd 2`rcd

  d n7.......4..7.........47........7

  n m7 B d ..7 Y n 4..7 Y m ..47  9d jaci'sa,' is $ ; kd ndsi s`' kds tris cdtramis d'tira`ris

  u'd cdtraz d'tasacNtramd (Kt B -(7 ji `rji' # is ji kd 2`rcd

  ...n7..........47

  -n..7 B n -4..7

  kd jaci'sa,' is 4 ; kd ndsi istd 2`rcdjd p`r kd cdtraz d'tira`r 

 _ri(ubtd ?&. ^isui"td i" '$*?'*$?'=. Ca rispuistd bOcir` '0&# 

Od$`s k`s vimt`ris u J 84 4 43 v J 8? 4 G43 9 J 84 4 ?3 $i 2!. dkkd sa is p`sa+ki ubvimt`r t tdk Dui Lu v tF sidb ubd +dsi $i 2!.Aôdha lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  lttp5""is.smranj.m`c"AôdhaMdrrdsmdk

^ispuistds $7

  C`z` C`z`

  Z' @i'a` is u' usudra` nrakkd'ti i' u'd mdti&`r*d.

  _drd Dui uLv ; t 2`rci' ndsiL jini' sir ka'idkci'ti a'jipi'jai'tis. a k` m`k`mdc`s m`c`

m`kuc'ds ji u'd cdtraz ik jitirca'd'ti risuktd'ti jini sir jasta't` ji mir`.

  _`r iOicpk` t B LL47

  ?ic`s Dui ik jitirca'd'ti5

  !.4.....!

  !.4..4...! B 4 jasta't` ji mir`

  !.4.-4..4.!

  dk sir ka'idkci'ti a'jipi'jai'tisL uL v ; ik t Dui ti li jdj` 2`rcd' ndsi.

 _ri(ubtd ?#. ^isui"td i" '=*?'*$?'=. Ca rispuistd bOcir` '0#

Ib C>83 m`bsa$irdc`s ik m`b&ubt` `rcd$` p`r kds cdtramis $i >>84 ?3 84 43 84 438? 43 8? 43 84 43. Hr`+dr Dui is ub m`b&ubt` kabidkcibti ab$ipib$aibti 9 dcpkadrk` d ubd +dsi$i C> 83 $i tdk `rcd Dui kds m``r$ibd$ds $i kd cdtraz $i >> 84 >! :3 ib $am1d +dsi sid 8:! > 43Aôdha lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  lttp5""is.smranj.m`c"AôdhaMdrrdsmdk

^ispuistds 47

Aôdha Mdrrdsmdk C`z` v4.5 ##"44"#4# lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  Aôdha Mdrrdsmdk

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Aôdha Mdrrdsmdk C`z`. Ispdma`s vimt`radkis. _r`nkicds ji Qdl``! ̂ ispuistds v4.: "#"#4$._%&a'd = ji 44

  C`z` C`z`

  Z' @i'a` is u' usudra` nrakkd'ti i' u'd mdti&`r*d.

  Jasp,' kds tris cdtramis Dui ti jd' m`c` m`kuc'ds ji u'd cdtraz. Ik rd'&` ji jamld cdtraz '`s

jd ik 'ucir` ji vimt`ris cdtramis i' tu mds`7 ka'idkci'ti a'jipi'jai'tis.

  4..4..47  ..4..47

  ....47

  4..4..47

  vic`s Dui ik rd'&` is $ 2*Odti Dui ik jitirca'd'ti ji kds tris pracirds 2akds is jasta't` ji mir`7L

 p`r k` Dui kds tris cdtramis m`'statu;i' u' m`'Ou't` ka'idkci'ti a'jipi'jai'ti.

  _drd kd si&u'jd pdrti ismranar%s ti k` v`; d p`'ir i' 2akds kd cdtramis ##L is jimarL kd cdtraz

  d..n7

  m..j7

  kd ipris` m`c` dL nL mL j7L T`h7

  4L #L $L :7 B : 4LLL47 Y $ 4L4LL47 Y # 4L4L4L47 Y 4 dL nL mL j7

  j`'ji dLnLmLj7 is kd cdtraz Dui 'imisatds pdrd dcpkadr kd ndsi  sucd vimt`ris ; risuikvi ik sasticd

  4 B : Y $ Y # Y dL ji j`'ji d B ->

  # B $ Y #Y nL ji j`'ji n B -$

  $ B # Y mL ji j`'ji m B4

  : B : Y $Y # Y jL p`r k` Dui j B -6

  ; kd cdtraz pijajd sir%5

  ->..-$7

  4...-67

 _ri(ubtd ?<. ^isui"td i" '=*?'*$?'=. Ca rispuistd bOcir` '0#& M`bsa$irdc`s k`s m`b&ubt s̀ !6 B4JL8? 4 ?3 84 4 G43 8> ? 43F B>JL8> 4 43 8 4 ?3 8G4 ?43F. @bd viz pr`+d$`s Dui dc+`s m`b&ubt`s s`b +dsis $i ! Em`c` $itircabdr ik m`b&ubt`$i vimt`ris $i ! Dui p`siib kds cascds m``r$ibd$ds rispimt` $i dc+ds +dsisKAôdha lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  lttp5""is.smranj.m`c"AôdhaMdrrdsmdk

^ispuistds 47

  C`z` C`z`

  Z' @i'a` is u' usudra` nrakkd'ti i' u'd mdti&`r*d.

  _r`ndj` Dui s`' ka'idkci'ti a'jipi'jai'tis ;d sdnisL pdrd mdjd &rup` ji vimt`risL m`k`md mdjd

vimt`r m`c` m`kuc'ds ji u'd cdtraz ; pruind Dui ik jitirca'd'ti is jasta't` ji mir`7.

  id L ;L z7 kds m` r̀ji'djds jik vimt`r pijaj`. acpkici'ti ldmic`s5  L 4L7 Y ; 4L4L-47 Y z #LL47 B #L4L47 Y ; 8L4L7 Y z -4LL47

  ; si risuikvi ik sasticd `pirds k`s vimt`ris i a&udkds m`cp`'i'ti d m`cp`'i'ti7

  ; Y # z B # Y 8 ; - z

  Y ; B Y ; puis vdki7

  -; Y z B Y z

  ji j`'ji ; B - ; sustatu;i'j` i' kd pracird imudma`'

  - Y # z B # - 8 - z

  $ z B -=

  kui&` z B - #

  9`s vimt`ris pijaj`s s`' ji kd 2`rcd5 L -L - #7. Z'd ndsi ji isti sunispdma` is5 4L -4L -#7

Aôdha Mdrrdsmdk C`z` v4.5 ##"44"#4# lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  Aôdha Mdrrdsmdk

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Aôdha Mdrrdsmdk C`z`. Ispdma`s vimt`radkis. _r`nkicds ji Qdl``! ̂ ispuistds v4.: "#"#4$._%&a'd ji 44

 _ri(ubtd ?1. ^isui"td i" '=*?'*$?'=. Ca rispuistd bOcir` '0## 

-id BJLu4....ubF ubd +dsi mudkDuaird $i ub ispdma` vimt`radk EMu"kis s`b kds m``r$ibd$ds$i k`s vimt`ris u& rispimt` $i BKAôdha lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  lttp5""is.smranj.m`c"AôdhaMdrrdsmdk

^ispuistds 47

  C`z` C`z`

  Z' @i'a` is u' usudra` nrakkd'ti i' u'd mdti&`r*d.

  9ds jik vimt`r u4 s`' 4L L L .... L 7

  kds jik vimt`r u# s`' L 4L .....L 7

  kds jik vimt`r u$ s`' L L 4L ...L 7

  ...

  kds jik vimt`r u' s`' L L L ...L 47

 _ri(ubtd '?. ^isui"td i" '=*?'*$?'=. Ca rispuistd bOcir` '0#1Mudb$` ti pa$ib ub m`b&ubt` _JL p83 i HT>83^] / p8?3Jdkd pZ843J+itd F EMýc` si 1dmiKAb`rcdmaýb d$ama`bdk6 Ik i&irmama` b`s pa$i 1dkkdr pdrd Dui vdk`ris $i dkd 9 +itd ib isim`b&ubt` 1dmib Dui sid ub ispdma` vimt`radk 9 pdrd k`s vdk`ris `+tiba$`s `+tibir ubd +dsi 9kd $acibsaýb $i $am1` su+ispdma`Aôdha lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  lttp5""is.smranj.m`c"AôdhaMdrrdsmdk

^ispuistds 47

  C`z` C`z`

  Z' @i'a` is u' usudra` nrakkd'ti i' u'd mdti&`r*d.

  a dk2d ; nitd s`' m`'std'tisL

  p 7 B d Y n  p X 7 B d

  sa p X 47 B nitd B dL d vdki nitd

  sa p 7 B dk2d B d W Y nL ji j`'ji n B dk2d

  p`r k` Dui p 7 B nitdW Y dk2d

  _`r k` Dui i' kd ndsi L 47L si trdtdr% ji k`s vimt`ris5

  nitdL 7 L L dk2d7

 _ri(ubtd ''. ^isui"td i" '=*?'*$?'=. Ca rispuistd bOcir̀ '0<$

Hui$` m`k`mdr p`r i&icpk` ub vimt`r 84?>!3 ib `rcd $i cdtraz 84 ?3 08> !3 $i >>K

Ab`rcdmaýb d$ama`bdk6 Y` Dui Duair` is 1dkkdr kds m``r$ibd$ds $i kd cdtraz isd m`b rispimt`dk su+ispdma` L4 ? ? 4FL? 4 4 ?FAôdha lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  lttp5""is.smranj.m`c"AôdhaMdrrdsmdk

^ispuistds $7

  C`z` C`z`

  Z' @i'a` is u' usudra` nrakkd'ti i' u'd mdti&`r*d.

  a. _uijis iprisdr ik vimt`r 4LL#L$7 ji kd 2`rcd

  4L7

  #L$7

  du'Dui k` '`rcdk is Dui k` ld&ds dk rivNs

  is jimar L iprisdr kd cdtraz

Aôdha Mdrrdsmdk C`z` v4.5 ##"44"#4# lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  Aôdha Mdrrdsmdk

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Aôdha Mdrrdsmdk C`z`. Ispdma`s vimt`radkis. _r`nkicds ji Qdl``! ̂ ispuistds v4.: "#"#4$._%&a'd > ji 44

  4L7

  #L$7

  m`c` vimt`r5 4LL#L$7

  d; u' as`c`r2asc` i'tri dcn`s.

  Cds nai' k` Dui pri&u'tdr*d is a'jamdr kds m``rji'djds ji kd cdtraz  4L7

  #L$7

  rispimt` ji kd ndsi 2`rcdjd p`r kds cdtramis

  4L7

  L47

  ;

  L47

  4L7

  Q is cds si'makk` iprisdrk` m`c` vimt`ris. Ist` is5

  4LL#L$7 B d 4LLL47 Y nL4L4L7

  ji j`'ji  4 B d

  B n

  # B n

  $ B d

  sasticd a'm`cpdtanki. Is jimarL '` puijis iprisdr kd pracird cdtraz m`c` m`cna'dma,' ka'idk ji

kds `trds j`s cdtramis.

 _ri(ubtd '$. ^isui"td i" '1*?'*$?'= d "ds ''20? )`rd ispd>`"d+. ^ispuistd bOcir` ''# 

EMdkmukdr $acibsa`bis 9 +dsi $i ispdma`s vimt`radkis "k,i+rd kabidkK

Aôdha lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  lttp5""is.smranj.m`c"AôdhaMdrrdsmdk

^ispuistds 47

  Aôdha Aôdha  Z' @i'a` is u' usudra` nrakkd'ti i' u'd mdti&`r*d.

  d7 9ds jaci'sa`'is ji I ; IX s`' rispimtavdci'ti # ; $L puis i' mdjd mds` kd cdtramis s`'

ka'idkci'ti a'jipi'jai'tis. up`'&` Dui sdnir vir Dui k` s`'5 m`k`mdr kds j`s cdtramis ji I m`c`

# 2akds ji u'd cdtrazL p`jr%s m`cpr`ndr Dui ik rd'&` ji istd cdtraz

  4....4..47

  ..4..4..47

  is #L p`r k` Dui s`' ka'idkci'ti a'jipi'jai'tis ; kd jaci'sa,' ji I is #.

  Jik casc` c`j` m`' IX

  4..-4..4..7

  4...4..4..47

  ...4....7

Aôdha Mdrrdsmdk C`z` v4.5 ##"44"#4# lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  Aôdha Mdrrdsmdk

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Aôdha Mdrrdsmdk C`z`. Ispdma`s vimt`radkis. _r`nkicds ji Qdl``! ̂ ispuistds v4.: "#"#4$._%&a'd 8 ji 44

  ik rd'&` is $ ; kd jaci'sa,'L $ tdcnaN'.

  n7 9` pracir` Dui ldmic`s dl`rd is `nti'ir kds imudma`'is ji mdjd u'` ji k`s sunispdma`s.

  ^imuirjd Dui kd jaci'sa,' ji u' sunispdma` is a&udk d kd jaci'sa,' jik ispdma` vimt`radk j`'ji

istdc`s trdndOd'j` : i' ik mds` ji kds cdtramis ##7 - ik ']cir` ji imudma`'is ka'idkci'tia'jipi'jai'tis Dui ji2a'i' ik sunispdma`. I' ik mds` ji IL ik 'ucir` ji imudma`'is sird : - # B #

imudma`'is. TM,c` si `ntai'i' (ôdji u'd 2akd ; z t

  4....4..47

  ..4..4..47

  ..;..z..t7

  ik rd'&` jini sir #. )`cdc`s j`s jitirca'd'tis $$ ; k` a&udkdc`s d mir`. _`r iOicpk`5

  z B Y ; ji kds tris pracirds m`kuc'ds7

  z B t ji kds tris uktacds7

  9` casc` m`' IX. (ôdji u'd 2akd ; z t ; m`c` ik rd'&` ji kd cdtraz risuktd'ti jini sir $L ik

jitirca'd'ti sird 'uk`5

  4..-4..4..7  4...4..4..47

  ...4....7

  ...;..z...t7

  `nti'ai'j` kd imudma,' z B

  A)I^IMMAV JI Z1I_(MAV. Jini vira2amdr sacukt%'idci'ti kds imudma`'is ji I ; ji

IX

  is jimar5

  z B Y ;

  z B t

  z B

  sa ris`kvic`s ; jiOdc`s i' 2u'ma,' ji

  z B t B z B ; B z - B - B

  u'd ndsi si `ntai'i dk p`'ir u' vimt`r &i'Nram`

  L ;L zL t7 B L L L 7 B 4L L 4L 47

  is jimar 4L L 4L 47L kd jaci'sa,' is 4

  ZC( JI ZI_(MAV

  jac Z a'tirsimma,' ?7 B jac Z Y jac ? - jac Z Y ?7

  4 B # Y $ - jac u Y ?7

  ji j`'ji jac Z Y ?7 B :

  dl`rd i' &i'irdk7 m`k`mds k s̀ vimt`ris ji I ; k`s ji IX m`c` m`kuc'ds ji u'd cdtrazL ; t`cdc`sjac I Y IX 7 B : i' 'uistr` mds` m`kuc'ds ka'idkci'ti a'jipi'jai'tis ji jamld cdtrazL Dui 2`rcd'

kd ndsi.

  p`r iOicpk` 4L4L4L7LL4LL7L4LL4L47LL4L4L47

  m7 kd sucd ji dcn`s is ik ispdma` vimt`radk ji kds cdtramis mudjrdjds ##L du'Dui '` is sucd

jarimtdL puis kd a'tirsimma,' ji dcn`s sunispdma`s '` is mir`.

 _ri(ubtd '=. ^isui"td i" $=*?'*$?'= d "ds ??20 )`rd ispd>`"d+. ̂ ispuistd bOcir` '0

Ci $db ub su+ispdma` vimt`radk $i T: m`b ik pr`$umt` ismdkdr iumk($i` ik pr`+kicd k`tib,` ib ist`6 @J LN9GzNtJ? >GzN!tJ?F $i d1( tib,` Dui sdmdr k`s vdk`ris pdrd Dui si ci

Dui$i m`c` ist`6 _JW84>K4?38?44?384?K>43 U. b` $a,` Dui si ci Dui$i a,udk puis s`b

Aôdha Mdrrdsmdk C`z` v4.5 ##"44"#4# lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  Aôdha Mdrrdsmdk

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Aôdha Mdrrdsmdk C`z`. Ispdma`s vimt`radkis. _r`nkicds ji Qdl``! ̂ ispuistds v4.: "#"#4$._%&a'd 4 ji 44

su+ispdma`s $astabt`s $a,` Dui si ci Dui$i m`b kd cascd `rcd Dui ik su+ispdma` _.Aôdha lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  lttp5""is.smranj.m`c"AôdhaMdrrdsmdk

^ispuistds #7

  Aôdha Aôdha

  Z' @i'a` is u' usudra` nrakkd'ti i' u'd mdti&`r*d.

  ?dc`sL Dui k` Dui Duairis is `nti'ir u'd ndsi d pdrtar ji kds imudma`'isL i' isti mds` mdrtisad'dsL

ji u' sunispdma`.

  1ai'L ti' i' mui'td Dui kd jaci'sa,' ji u' sunispdma` is a&udk d kd jaci'sa,' jik ispdma`

vimt`radk j`'ji istdc`s trdndOd'j` ci'`s ik ']cir` ji imudma`'is ` ristramma`'is Dui ji2a'i' ik

sunispdma`.

  I' tu mds`

  jac 3 B jac I - '/ im. B : - # B #

  Jinis ris`kvir ik sasticdL ji j s̀ imudma`'is m`' mudtr` a'm,&'atds. Jinis p`'ir : -# B #

 pdr%citr`s.  Y;-zYtBL

  #-zY$tB

  sa ristdc`s - ; Y # t B

  ; B Y # t v`; d jiOdr d ; t m`c` pdr%citr`sL ; jispiOdri ik rist` ji kds a'm,&'atds i'

2u'ma,' ji jaml`s pdr%citr`s7

  z B Y ; Y t B Y Y # t Y t B # Y $ t

  dl`rd p`'ic`s u' vimt`r &i'Nram`

  L ;L zL t7 B L Y # tL # Y $ tL t7

  ; sdmdc`s 2dmt`r m`c]' d k`s pdr%citr`s

  4L 4L #L 7 Y t L #L $L 47

  kui&` kd ndsiL istdr*d 2`rcdjd p`r k`s j`s vimt`ris d'tira`ris5  1 B U 4L 4L #L 7 L L #L $L 47\

 _ri(ubtd '0. ^isui"td i" $=*?'*$?'= d "ds $?2' )`rd ispd>`"d+. ̂ ispuistd bOcir` '#1

EMu"kis $i k`s sa,uaibtis m`b&ubt`s $i vimt`ris ib : s`b kabidkcibti $ipib$aibtisKd3 8 4 > 4 G> 3 8 ? G> G> ? 3 8 ? > ! 4 3 8 ! ? G! ; 3+3 8 : : ? ? 3 8 ? ? ; ; 3 8 G\ ? \ \ 3

Aôdha lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  lttp5""is.smranj.m`c"AôdhaMdrrdsmdk

^ispuistds 47

  Aôdha Aôdha

  Z' @i'a` is u' usudra` nrakkd'ti i' u'd mdti&`r*d.

  a m`k`mds k`s vimt`ris m`c` 2akds ` m`kuc'ds7 ji u'd cdtrazL ik rd'&` ji kd cascd '`s jdrd ik

'ucir` ji vimt`ris ka'idkci'ti a'jipi'jai'tis.

  d7 4 L # L 4 L -# 7

  ... L -# L -# L 7

  .. L # L $ L 4 7

  .. $ L L -$ L = 7

  ldz mir`s p`r @duss ristd'j` 2akds ; t`j` is` Dui sup`'&` Dui m`'`mir%s7

  +$Y+# +:-$+4L itm. ?ic`s Dui ik rd'&` is : ` sa Duairis ldz ik jitirca'd'ti ; m`cpruind Dui

is jasta't` ji mir`7. 9ui&` k`s : vimt`ris s`' ka'idkci'ti a'jipi'jai'tis.

  n7 : L : L L 7 L

Aôdha Mdrrdsmdk C`z` v4.5 ##"44"#4# lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  Aôdha Mdrrdsmdk

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Aôdha Mdrrdsmdk C`z`. Ispdma`s vimt`radkis. _r`nkicds ji Qdl``! ̂ ispuistds v4.: "#"#4$._%&a'd 44 ji 44

  ... L L = L = 7

  ... -6 L L 6 L 6 7

  p`jic`s m`cpr`ndr Dui tdcnaN' ik rd'&` is $L p`r k` Dui k`s tris vimt`ris s`' ka'idkci'ti

a'jipi'jai'tis 2aOdti Dui ik jitirca'd'ti 2`rcdj` p`r kds tris pracirds m`kuc'ds is jasta't` ji

mir`7.

 _ri(ubtd '. ^isui"td i" $=*?'*$?'= d "ds $?2$$ )`rd ispd>`"d+. ̂ ispuistd bOcir` '<?

-id @ ik su+ispdma` $i ! $iaba$` acpk(matdcibti p`r kd imudmaýb N 9 K z J ?.Q ^?.\ pubt`s] dkki ubd +dsi $i @.Q ^?.\ pubt`s] Itaib$d kd +dsi mdkmukd$d d ubd +dsi $i !.Yd pracird pdrti 9d kd tib,` mdkmukd$d6 Bdsi @ J L8G4 4 ?38G4 ? G43F 8mri` Dui ist" +aib3.Ek,uaib sd+i m`c` si 1dmi ik si,ub$` pubt` $ik i&irmama`K.Aôdha lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  lttp5""is.smranj.m`c"AôdhaMdrrdsmdk

^ispuistds 47  Aôdha Aôdha

  Z' @i'a` is u' usudra` nrakkd'ti i' u'd mdti&`r*d.

  acpkici'ti dôdji u' tirmir vimt`r Dui sid ka'idkci'ti a'jipi'jai'ti m`' k`s jdj`sL ̀ k` Dui is

k` casc`L u' vimt`r Dui V MZC_9( kd imudma,' jik sunispdma`5 Y ; - z B L is jimar z B Y ;

  p`r iOicpk` ik 4L 4L $7 L vis Dui z '` is a&udk d Y ;.

  9d ndsi iti'jajd sir*d U-4L4L7L-4LL-47L 4L4L$7\

  puijis m`cpr`ndr Dui istd nai' sa' cds Dui ldmir ik jitirca'd'ti ji kd cdtraz Dui tai'i p`r

m`kuc'ds k`s vimt`ris d'tira`ris ; m`cpr`nd'j` Dui is jasta't` ji mir` pdrd Dui sid' ka'idkci'ti

a'jipi'jai'tis ; 2`rci' ndsi7

 _ri(ubtd '&. ^isui"td i" $=*?'*$?'= d "ds $'2? )`rd ispd>`"d+. ̂ ispuistd bOcir` '<$Od$` ik su+m`b&ubt` 2JL8 9 z36 T! N 9T! N zT! J 4F m`cpr`+dr sa si trdtd $i ubsu+ispdma` vimt`radk $i T!.Hri,ubtd6 EBdstd m`b $imar Dui m`c` ik vimt`r buk` $i T! GU?J8? ? ?3 b` mucpki kdimudmaýb b` is ub su+ispdma` $i T!K -a is ds( b` dmd+` $i ibtib$ir ik p`rDui sa dk,uaib cik` pui$i ipkamdr.Aôdha lttp5""333.mdrrdsmdk.'it:=.'it"2*samds"  2*samds0;dl``.is  lttp5""is.smranj.m`c"AôdhaMdrrdsmdk

^ispuistds $7

  Aôdha Aôdha

  Z' @i'a` is u' usudra` nrakkd'ti i' u'd mdti&`r*d.

  _drd Dui sid u' sun7ispdma` vimt`radk jini mucpkarsi Dui sa t`c` u' vimt`r jik sunispdma` ;

`tr` vimt`r jik casc`L su sucd jini pirti'imir dk sunispdma`.

  _`r iOicpk` t`cic`s ik 4LL7 Dui mkdrdci'ti pirti'imi dk sunispdma` puis 4K$ YK$YK$B4

  ; ik L4L7 Dui tdcnaN' pirti'imi dk sunispdma` puis tdcnaN' vira2amd kd imudma,'

  _uis nai'L su sucd 4L4L7 V pirti'imi dk sunispdma` puis 4K$Y4K$YK$ '` is 4.

  _`r k` Dui V I Z Z1I_(MAV ?IM)V^A(9 ji ^$.

  (jicds jinir*d ji vira2amdr Dui sa is u' vimt`r Dui pirti'imi dk sunispdma` ; k` cuktapkamds p`r

u' 'ucir` kd'jd pirti'imai'ti dk muirp` i' ik Dui istic`s trdndOd'j`L kd'jd W tdcnaN' jini

 pirti'imir dk sunispdma`.

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