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Y68]^SP24&68<KORZ._F>687`?EOE68<K<>682aC(FK7`?=F>6bQ@A&7(9
687cIJd&BeA&2&BZ
fgGhRikjml H+S;?EF>?E687=BXNP2&Q@A&2&B?E?WSPO=QonpZ`qrarRspt0spu v , . . . , v r@v;wxt$s w , . . . , w q!y>spz{spu^|3s~}Xspr K 
1
r
1
s
€s‚o|0v;ƒpƒ=„;…†z{s‚r V ‡ qrˆXs8y‰|3s
l YŠ‹n v , . . . , v r‚tŒu.}y‰tus=„;ƒ[…Žu.„X‚G‘;u†ˆ;tKˆ’…†u}
1
r
l Y^“n Lin(v
Z
1 , . . . , v r , w1 , . . . , w s ) = V
”„;yŒy•r–}„Pr r — |J…R˜spy (v , . . . , v ) u^tJ™pŽ|xWspƒ=s8tŒ|ŒrDsptŒusšT„Prpt›rœov;u V t›r8|ŒžŸ;„;uu s‚rb};…Žƒc™p{¡–tŒuŽ¢@…†u.„; —
z{sTˆ†sEsptKˆ;us8|3spƒ~€s‚o|01veƒ=spu£„e…Xrr (w , . . . , w ) ¢@…¤sptuspƒšT„Pr‚trbœov;u V spƒ$ˆ‘;uŽ¢R|ws8ƒc}Xs8u ‡
1
s
šTs8wspt›r ‡Ž¥ FK7IJd&]&7E682M4&682’H+68C68F>?+4GA&7E\W]M¦:;<K<>?=OWNP24GFKB6–1324GA&§ŽO=F>:;25Y68Q@d&Be<KF>\E] s ZX1¨?=O (v , . . . , v )
Y687E68FKOE?D68FK26{H(S;?=F>?b¦:;2 V © ?W:M?E68O=QR682£C(FK7 s := 0 l 1324GA&§ŽO=F>:;2?WSP2GI0SP2&B†npªFK2«4GF>6@?E68¬­1U^SP<K<>6{BeFKr<KO
4GF>6[®TA?E?WSPB6eZ&¯TA&2°?E68F4GF>6[®TA?E?WSPB6IJd&7 s = m ∈ {0} ∪ N 7=F>\E]†O=FKB l 1-24GA&§†O=F>:;2?‚SP2&2^SP]&¬±6‹npZ².A
QR68FKB682aF>?=O © 4S;?W?(?=F>6b4SP2&2¤SPA\E]°IŒd&7 s = m + 1 BeFK<KO l 1-24GA&§†O=F>:;2?=Y68]^SPA&V&O=A&2&B†npZ
³
?`?E68F>682 SP<>?E: w , . . . , w ∈ V A&24M268Y682 l YŠ‹n!B68<KOE6
m+1
1
l Y^“†´Pn
Lin(v1 , . . . , vr , w1 , . . . , wm+1 ) = V
Z
U^SP<K<>?
2&F>\W]XO{H(S;?=F>?µ¦:;2 F>?=O BeFK<KO
ª¶C!68B682 l Y^“†´Pn·F>?=O±SP<>?E:
r) $ V
¬FK24&6@(v?=OE168,2.?. .68,FKv2 r )w l C:;Y68F j ∈ {1,V. . . , m© + 1}Lin(v
n·2&F>\E1]†,O. .FK2. , vLin(v
682XO=]^SP<KOE682Z FK7
QR68FKB682 © 4S;?W? v , . . .j , v , w <KFK26RSP7A&2^SPY&]^NP2&BeFKBµ?=FK24Z†¸bSPQ@A{B68<KOE6 λ 1v, .+. . ·, ·v·r+) λ v + µw ¥ = o
¬FKOxB68C(F>?E?E6825L†1 §eSP<›SP7E682 r λ j, . . . , λ , µ Z ¥ NP7E6 µ 6= 0 © 4SP2&2±C!NP7E6 w 1∈ 1Lin(v , . . .r, vr ) © C+S;j?
4&687
®2&2^SP]&¬±6·d&Y687 w C(F>4&6817E?=V&7=F>\W]XORrZ®T<>?E:F>?=O µ = 0 A&24°?W:;¬FKO λ vj + · · · +1 λ v =r o Z ¥ 68B682
l YŠ‹nTI$:;<KBeOb]&F>687WSPA? j λ = · · · = λ = 0 Z¹LŽ:;¬FKOD?=FK244GF>69
68§ŽOE:;17E6812 v , . . . , vr , wr <KFK26RSP7[A&2Gº
SPY&]^NP2&BeFKB © §»;2&2682M2^S;1\E]a1-24GA&§†O=F>:;r2?‚SP2&2^SP]&¬±6[SP<>?E:D4GA&7E\W]a_FK2&Q@A&2^SP]&¬±1 6–B6@68FKBer268OEj6879
68§†OE:;7E682
SPA?`4&68¬ mº0O=A&V68< (w , . . . , w , w , . . . , w ) Q@A°68FK2687H(S;?=F>?(¦:;2 V 687=BXNP2&Q@OC!687E4&682Z
LŽ:;¬FKO`BeFK<KO4GF>6[®TA?E?WSPB16[SPA\W]MIJj−1
d&7 s =j+1m + 1 Z m+1
fgGhRi~¼½l ®A?=OWSPA?W\E]&<>68¬¬{SnpZqr[rRspt
¾s8tŒ|3spƒ[r@spt
B := (v1 , . . . , vn )
w = λ 1 v1 + · · · + λ k vk + · · · + λ n vn
¿·„euu¤trp|!„;…&™p
zµtJ|
sptŒusbšT„Pr‚trD}Xs‚r K  €s‚o|0v;ƒpƒ=„;…†z{s‚r
λk 6= 0.
(∗)
B 0 := (v1 , . . . , vk−1 , w, vk+1 , . . . , vn )
sptŒus[šT„Pr‚tr·œov;u ž!}  ž
ˆXs8|0„;…†r8™8†|!wspƒc}†spVu ‡ ‡ ‡
vk
P„;uu  …Žu^|3spƒ[q!ƒ‚Ž„;yÀ|J…†u†ˆ°}Xspƒ[š„Prpt›rRsptKˆXspu&r@™pŽ„cÁ8|  ˆXs$ˆ†spu
‡
V
w
„e…Xr —
Tš s8wspt›r ‡( ]&26¤H+6@?E\W]&7WNP2&§†A&2&B4&687î<K<KB68¬±68FK2&]68FKO l :&ZÄH–ZÄ4ZÄ®DZÅn±§;SP2&2SP2&B682:;¬¬±682C687E4&682 ©
4S;?E? k = 1 © SP<>?E: λ 6= 0 BeFK<KORZ
1
Š
l 1cn FK7QR68FKB682 4S;?E?
¸bSPQ@¥ AaB68<KOE6 ©
B 0 = (w, v2 , . . . , vn )
µw + µ2 v2 + · · · + µn vn = o,
Ç°FKO l ∗nI$:;<KBeO
C:;Y68F
SPA?<KFK26RSP7A&2^SPY&]^NP2&BeFKB682³x<>68¬±682XOE682ÆY6@?EOE68]XORZ
µ, µ2 , . . . , µn ∈ K.
µλ1 v1 + (µλ2 + µ2 )v2 + · · · + (µλn + µn )vn = o,
PS <>?E: l 4S B 68FK26µH+S;?EF>?¦:;2 V F>?=O‚n µλ = µλ + µ = · · · = µλ + µ = 0 Z ¥ 68B682 λ 6= 0
n
n
1
F>?=O µ = 0 A&24a4SP]687TSPA\W] µ = · · · =1µ = 02Z 2
l 1E1=nx¯TA&2’QR68FKB682ÃC(FK7 Lin(w, v2 , . . . , v ) =n V ZŽ³
?(?E68F v ∈ V B68B68Y682ZG¸bS B H(S;?=F>?!¦:;2 V F>?=O ©
6pȎF>?EO=F>687E682 µ , . . . , µ ∈ K © ?E:2 4S;?E? vn= µ v + µ v + · · · + µ v Z&¯S;\W] l ∗n!F>?=O
1
n
1 1
2 2
n n
v1 =
A&24a4SP]687
v=
1
λ2
λn
w − v2 − · · · −
vn
λ1
λ1
λ1
µ1
µ1 λn µ1 λ2 v2 + · · · + µn −
vn .
w + µ2 −
λ1
λ1
λ1
ŽL :;¬FKO`F>?=O v ∈ Lin(w, v , . . . , v ) Z
®A? l 1cnA&24 l 1W1cn
I$:;<KBeO © 24S;?E? Bn0 68FK26·H+S;?EF>?(¦:;2
V
F>?EORZ
fgGhRi[É5l ®A?EOWSPA?E\E]?WSPO=QonpZ;qrxr@s8t$spu (v , . . . , v ) …Žu.} (w , . . . , w ) šT„Pr@spu{}†s‚r K  €spo|0v;ƒpƒc„;…Žz{s‚r
¿D„;uu~ˆ;t08|[s‚rµ¢8…{Ê@s=}†spz v sptu w1 ž(r@vË}n„PrWr5„e…Xr (v1 , . . . , vm) wxt0s=}XspƒMsptusµšT„Prpt›r±œoveu V
V‡
1
n
spu^|Œr8|3s‚†|Œžwspu^u v };…Žƒc™8 w s8iƒWrRs8|›¢@|wxj tƒ=} ‡
i
j
šTs8wspt›r ³
??E68F68FK2 B68C!NP]&<KORZo¸–SP2&2·BeFKY&O¹6@?68FK2 w © 4S;?2&F>\E]XOŸQ@A Lin(v , . . . , v , v , . . . , v )
j
1
i−1 i+1
n
B68]»;7=ORª^‡ SP24&687=2GI0SP<K<>v?i C+NP7E6
Lin(w1 , . . . , wm ) ⊆ Lin(v1 , . . . , vi−1 , vi+1 , . . . , vn ),
C!S;?{C68B682 v ∈ Lin(w , . . . , w ) 4&687<KFK26RSP7E682ÍÌ2^SPY&]^NP2&BeFKBe§68FKOÃ4&6879
68§ŽOE:;7E682
C(F>4&687E?=V&7WN;\E]6eiZLŽ:;¬FKO?=FK214M4GF>6[9
m68§†OE:;7E682
v1 , . . . , v n
v1 , . . . , vi−1 , wj , vi+1 , . . . , vn
K< FK26RSP7–A&2^SPY&]^NP2&BeFKB l ?RZ.H+68¬±687=§ŽA&2&Ba“54&6879
:;7=<>6@?=A&2&B†npZ ¥ 68B682
O=d&7=<KF>\E]
Lin(v1 , . . . , vi−1 , wj , vi+1 , . . . , vn , vi ) = V.
Lin(v1 , . . . , vn ) = V
l ∗∗n
BeFK<KO–2^Soº
4GF>6Ë9
68§ŽOE:;7E682ÎFK2 l n5§68FK26H(S;?=F>?æ:;2
4SP2&2Χ»;2&2XOE682Ï?=F>6Ë2^S;\E]ÐLGSPO=Qъ~4GA&7E\W]
¥_FK2&NP7EQ@68A&2Î
2^SP]&¬±6D¦:;2 v Q@A°68FK2∗687H+S;?EF>?687=BXNP2&Q@OC!687EV4&68© 2Z.¸F>6@?F>?=OTSPY687`2&F>\E]†OT¬±»;Be<KF>\W] © 4S w ∈
BeFK<KO i© SP<>?E:4GF>69
68§ŽOE:;7E682 v , . . . , v , w <KFK26RSP7SPY&]^NP2&BeFKB{?EFK24ZŽU^:;<KBe<KF>\E]MY&FK<>4&j 682
Lin(v1 , . . . , vn )
n
j
4GF>6b9
68§ŽOE:;7E682aFK2 l ∗∗ n(68FK26[H(S;?=F>?+¦:;2 V Z 1
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