:<;>=@?BADC ED= FHGHIKJDLMGON,EP=RQ
STVUXWZY\[]Y\^`_aYbTVcXdfehgi"j8klfmXnokqpsrkut&iwvurkx1yVi"j{z|kqpayV}~7z|y|psiwXkt@naiwvurkrupsiZXk| psmi"jrku}i"j8kl2}"k,kunk|ju
j2~naXkyVmXz|yVtpsrkunsrk2g`kpayV}~7z|y|psiwk|ju}"yVnXV ht@i"jrupsiZXk}irk2g3y\1k|zuZix&k|ju2makyVm7j3j2Xz|Xiwkl
r|j8yVmr|kqXk|jju~8juvut k|j}inoir|k|jqk|z&yVmj2njprt vu\rk|j&¡¢£7¤¥¡§¦a¨¤H©«ª¬¨¯®a°©/©V°¡/±§²\¡©³´¬¨/³¥´/°µ7¢µ©
¡V¶7³H¨©7µ¬¡©¸·4®a¥©,²b®¹©7¡¬ª7¢°¨»ºs
¼ ^½d¾¿YZ^4À8UÁT¬cd2d2cfY\Â7Y_aÃ`ÄÅdÆ`dÀ1Ç[ÈcdW\YZ^`UX_YZT¬cXdɸÂÊOc7ËÌÂ7YbcX^4ÀTVUÁcXWZdÍH[]_aYZdWZ_BÎÏ_aÐÊs^¹ÑcÒT¬ÊHÂUXÑcÁT
cXdÀÀ8ÊsÃ`ÀÉBÎÏ_aYÀÓsUX^`UÁTV_WZc(Íoc7ÀÀ8ÊsÃd3WbcXd3T¬UXd1ÃWbÀ1_À¬dÊ(ÔÀ8cX^OÃdd2Ês^4ÀqcX^ÂÊ(TVcÖÕHT¬_aY\dÒ¯ÊsÃ`T&WZcXd3dÆ`duÀ8ÇX[]cXd
WZY\^`UÁ_aYbTVcXdÉÈÂÊOc7ËÌÂ7YbcX^4À¬d&ÂÊs[]ÒWZc7ÄHcXd×
.¯ /ØX'7Ù12Ú)ÛÜÝ 2ÙÛ Û2hÞX#2ß@2à "2" 2áÛâ4u¿| " "/'uÛ "Ù!ßÖ2ã
*{/"¥sÙ ä2 å'¬$s1äÙ&ÛOæç27 Ù" "7¥ è/¥&ß$ ÛOæéä2¯ØÁOæéàß@ÙÛh ä2ã
êáÙ2 "Ú@Û3ÛÙ&«4Ès2 Û«Ù3ÛX2Ú ÙÛ ëÞX#2ß@@ "2" qÛ%8ì52ØÁa/"Ù
ÝB%8í5"äÙÁ2'oÙî,HÙ)Ûu2Ú2ä " q, Ù! ß@ß@"ïbÙ ß V"ØÁØÁs2 ß@ÝÖ{Ù Ûa/2
Û ÙÛ , `ÞX#2ß@ãHð) ÙÛ Û2ÞX#2ß@2)Û #2â! Û" "¸ Ù$ "#ßÖ3äÙ /Á
Ûa/à$s1äÙÛoæñ "ä1/"ÙáÛàßÖ/ß V"ØÁ2ãÁ+ s2àÛÙä"ß@4Ù |7Û)äÙß@ Û ä2Ùß@ß@27Û
2 "ÞX#ßÖs227ò 3 2Ù "2ã
ó ô<õÅöÏ÷øOùûú÷ýü@õÅöõfþ øOÿ øHù ÷2þ ÷
+ïbÙ ßÖ " 2ß@271'OÖØÁa/Ùë2q "1 á "23/ "$ 2)Þë 27qÛÖßÖ#2 &2a/ 2
ÙVÜÙ 3Ý åä2f%ãoëÞX#2ß@ ""1 '4á4 "ÞX#ßÖ&ÛoæéØX/Ù3 "1 /'s2
Þ#ßÖÛ " "2ØÁa/"Ù2ãa 2ß@ "/'a "à Ù"ÞX#2ß@2â/7ÙÁâÛÞX#ßÖ "2" 2)e
3x + 6y = −3
−2x + y = 12
x + 2y + 3z = 5 2x + y − 4z = 0
2x 1 + 4x 2 = 0 3x 1 + x 2 = 0
−2x 1 + 2x 2 = 0
%
H{ 2ß@" 2 DÞ#ßÖ¸Û{Û 2ØÁa/"Ù ÝÈÛ "äÙÁ
Ù/2
x
uy
u'¯ ,Û"#ßÖ, Þ#ßÖÛÛ2 2ØÁaV"ÙÝ Ùhä2ÙX2
x
'y
z
h â Ù#2ß@ÞX#ßÖÛ Ù"h2ØÁa/"Ù Ý&ÛB"äÙÁ
Ù2
x 1
x 2
f "2Û
x
y
uã.¯, /ä|'( "âÞX#ßÖ/7âÙ7âaÛ2âÞX#2ß@2â "1/" )e
x − y 2 = 1 2x + 3y = 8
x + 3y = 5 2xy = 4
.¯]ou1'`Ûa @ÞX#ßÖÖÛ!7ä|'Ûa å 2ß@"#2 2ØÁa/"ÙO'HÖÛ2qä2ÙÁ2a/ ¿ÝB
"ä 3ØÁ
1
ã()â "3Þ#ßÖÛÛ Ù¿'aÛâ å&Û2#2ß@ØÁa/ÙO'a 2â"äÙÁ2âOæéa¬"27âaÛ3ßÖ#2 a 2Vã
"!#%$&"!$&'(#)'+*-,#)./!'102",3346587:9;<4=">)>&,?$@BA,!CD4=3'9EF#
x y x y z
ØÁaÛ{" OÞf{ Ù"ã(*{âäÙß@ßÖÁâ 2Ù2 â2æç" OÞBf%8í5&"äÙÁHGIÙ ""Ùf å& ßÖ 2
ÛqÙ "Ö 2ß@"#2 qä2ÙX
x 1
'4 "ÖÛ"#ßÖ
x 2
"
k
#2ß@x k
ãO.¯§Ù7 ""ä2ÙÁ2'4 "2
ÛB 2ß@"2 ÞX#2ß@2âÛ
%ÛX"27 s2ä2ß@27qe
3x 1 + 6x 2 = −3
−2x 1 + x 2 = 12
u
x 1 + 2x 2 + 3x 3 = 5 2x 1 + x 2 − 4x 3 = 0
JUK^YÀ¬YZÊs^ÅeXÙ27
n
p
Ûà27"2 ÙÁ "ãâ,ju~¬juvut k}inoir|kÛn
2ØÁaV"ÙÝp
"äÙÁ2BÞX#2ß@ÛBÞXse
a 11 x 1 + a 12 x 2 + · · · + a 1p x p = b 1
a 21 x 1 + a 22 x 2 + · · · + a 2p x p = b 2
ããã ããã ããã
a n1 x 1 + a n2 x 2 + · · · + a np x p = b n
-:
Ùîf 2ä2ÙX"L@ä227
a ij
u
b i
ÙÁâÛ 2 "-M 2ã
`3inoz|yVnanam(k|j3Ù7x 1
'
x 2
'Hã¿ã¿ã'
x p
ã
" 2 "a ij
Ù7f "]z|y8kONzuiwkunabj¸m]psrkut&iwkur{t kut@2rkuãPH§ 2ß@" "Û"ä¸"Û"ØÁ¸ "] "!'h "
Û"#ßÖ3 åäÙ "Ùã
` 2 "b i
Ù7â "2qz|y8kONzuiwkunajmj8k|z|yVnsÖt kut@2rkuãQ¯æéÛä2"Û"ØÁ " "!ã
RBSkutps}"keasÙ "ÞX#ßÖ
3x 1 + 6x 2 = −3
−2x 1 + x 2 = 12
í
ÙB
a 11 = 3
'a 12 = 6
'a 21 = −2
'a 22 = 1
'b 1 = −3
ub 2 = 12
ãJUK^YÀ¬YZÊs^0eÖÙ Ù]Û]ÞX#2ß@ "1
-:3&Å "uqÛÖ 2 "
(x 1 , x 2 , ..., x p )
ØÁá TMÙ "2ØX/ÙÛBÞX#2ß@ã4ð),ÛØÁOæç{ÞX#ßÖ2&z|yVtp\iZ2}"k3æé HBßÖÙàÙ ""ÙOã
RBSkutps}"ke
(−5, 2)
3qÙ Ù{Û§ÞX#ßÖ
íuã4.¯§"42'osÙ
x 1 = −5
x 2 = 2
'4Ù« $"3x 1 + 6x 2 = −3
−2x 1 + x 2 = 12
ãÁ.¯@ /ä|'(1, −1)
Oæç2a/Ù ÙÛ
í¥ä2 á4Ù
x 1 = 1
u
x 2 = −1
'a "Û2#2ß@32ØÁa/"ÙfOæç2a/â/ïZ¿/ãU VWøYXõù"Xþ ZÅöõfþ øOÿ øHù ÷2þ ÷
JUK^YÀ¬YZÊs^¹eaÛ2ÞX#2ß@2 "2" 2àÙ7&2miV}"kunaj2æç" àÙ7â "ß@ò2ß@22ß$ 3Û3Ù Ù2ã
[|j8yVmrk&ÞX#2ß@à "2" '8äæç2¯2qÛ2 ß@"2 Ù2 2Ù Ù2ã/Ù ¥äïZ" '¬Ùq ïbÙ ßÖ
hÞ#ßÖàå 2qÞX#2ß@à2ØÁ¿/ "27 ""ß@ "/'V"2ÞX#2ß@à2ä2Ù ¥"ß@ "/'ÜØÁOæéÝ
$sÙ" áÝÖÞX#2ß@ØÁOæçÙ ä|â ÙÛ ãH2 "ä"a 2¯Ùs2 /Ù¯ØÁ(s2 ß@ÁÛ ïbÙ ß@2
BÞX#2ß@ ""1 ,,Þ#ßÖ "1 2ØÁ/ 27ÙÁâ 2¿/723e
ß " " h "!)a äÙ|7ÙBÁ " ¬ÜÙ2λ
ïbÙà å& "!k
Ý å "!i
'aÙîλ
2â, 2 HØX2 "äÙØÁi 6= k
2ä|a!2 å& "!i
â å& "!k
72ã4ðfs2¯e
¬ÜÙ2 3Ýf "!@äÙß$aÙ§Û 23 "!
ßÖ åä2 ) "^]Á#2ß@& ""!
L k
a
L k + P
i6=k λ i L i
uã_\2 "& X27Ý&ïZ/" "" Ùs2 /Ù2 2ß@27|/" àä2ä22ã
"ß@2 2 "!0 = 0
ä2Ùä2 ØÁ )ÞX#2ß@3OæñÛÙ ""Ùh2æç" 4ä2Ùß@4Ù "!âÛÞX40 = b i
12ä
b i 6= 0
ã) 2ßÖ 2Û3 Ù ÙÛÞX#2ß@2â "1 àä2ÛÙ2'aÙf" " ä2ÙXÛa!/7e
L i ← λL i
Û &ØÁ@ å"#ßÖ "!qÛ]Ù2]Þ#ßÖ@2q!7 "@Ý
λ
ïbÙ" ""#2ß@& "!Û \æñä2"fÞX#2ß@ãL i ← L i + λL k
Û" ØÁ "qÙ2 ") "!âo!7 "Ý wæéä"2 "!
+ λ
ïbÙ"h wæéä22 "!k
L i ↔ L k
2âÛ 3ØÁ3 "2 "!2
i
k
Ù7âä|a!22`Pa+b cedfgihkjlfnmkfpo qYrstjluwvxlstyzm|{}u~yr-rvgfjlx+y~qYtxlo f
\Ù"Û Ù ÞX#2ß@/7e
3x + 6y = −3
−2x + y = 12
}
.¯Bß " å7h å 2ß@"# "!)
1/3
'ÙfÙ$"27 ÞX#2ß@ØX/ "73eL 1 ← 1 3 L 1
L 2
x + 2y = −1
−2x + y = 12
ì:
¯â{8ÜÙ|/ÁÛ2fïbÙ å 2ß@"# "!Ý& "2äÙÛ "!'ÙfÙ$"273e
L 1
L 2 ← L 2 + 2L 1
x + 2y = −1 5y = 10
&
.¯BÛX"7 åÛ"#ßÖ3 "!)
5
'ÙfÙ$"7L 1
L 2 ← 1 5 L 2
x + 2y = −1
y = 2
6:
.¯ Mao's,Ù 1Þ7ÛïbÙ åÛ"#ßÖ3 "!Ý& " 2ß@"# "!/'ÙfÙ$"7e
L 1 ← L 1 − 2L 2
L 2
x = −5
y = 2
9
H3ÞX#2ß@q&ÛÙä33Ù ÙØÁe
(−5, 2)
`Pa)` qYrstjluwvxlstyfYyvoQBtxljljlyvm~xlo fY_vfgfYyvru~oj+fv (kj+fYtumkfYrs-f%xlfYyvr
HÙ ØÁOæéÙ 1/ " " àÛàÞX#2ß@2Û! Û" "'7 2äÙ" àÝqä|ØÁïbÙ "âÙß<Ûà"äÙÁ
X¿fïZ/!a71ã¯w' "ÖßÖ/2ßÖ/ä2"2'¯ØÁâÙÁ Û2Öò 2Öa 224'à ïb# Á " 1/ " "
Û äßÖ7 ")$ "2fÛ2äÙLÖä"Á2ã(f "2fÛOæç2ä &e
3x + 6y = −3
−2x + y = 12
3 6 −3
−2 1 12
&
H2äÙL@ä2"7Ûf 2ß@"2 hßÖß$ a/ 7âÝ!7ä|Û) åq$ )2 ä1 'u "àä2ÙX"L@ä"27âÛ
2ä2ÙÛ§ßÖß$ qÝ@Û Ù¿/ã`@ 2Ù ""Ù,qïZâäÙßÖß@3 ä2Û2ß@ßÖ71'oï¯ØÁOæçÙ¸q2æç2ßq$4òu Ý
2ä2Ù" "2ä2ÙX22ãsð)BÙ$"27ä2ä22ß@27&e
L 1 ← 1 3 L 1
L 2
1 2 −1
−2 1 12
L 1
L 2 ← L 2 + 2L 1
1 2 −1 0 5 10
L 1
L 2 ← 1 5 L 2
1 2 −1 0 1 2
u Ma
L 1 ← L 1 − 2L 2
L 2
1 0 −5 0 1 2
ä/æéÝÛ" ÞX#2ß@Öe
x = −5
y = 2
%85
ð)Bä2Ùä "äÙß@ßÖ) 2ä2ÛßÖß@27ØÁ3 "ÞX#2ß@"ØÁ3Ù ""Ù§e
(−5, 2)
ã) "¿'aÙf 1/" " qÛ äßÖ7 ")$ 1fÛäÙLÖä"Á2ã
`Pa} Pstj+uvx+sty~rm~fPukfYjlPu~fYrr-rvgfYrjlx+y~qYtxlo fYrrx+gihjlfYra
`/ÁÛÛÙ Ûoæñ ?ßÖ "2âÛ Ù ÙÛqÞX#ßÖ "2" 22'Ùä2`ØÁ2 ØX2?2ß@ "
ÛfÞX43Û3Þ#ßÖ)ØÁ HÙBÙ3$4Ù 2ã
ÄHcX[]ÒWbcO×T O×&¡¢HÞX#2ß@
x + y = 3 0 = 5
%%
Oæéqa/àÛ)Ù ""Ù2'Áä1 h åÛ2#2ß@)2ØÁa/"Ù )s2aàòu /"ïZ//ãXqæé)ßÖ"# â!22 / "'X
Ù âÛ "@ 2Ù ""Ù,ÛOæé§ÞX#2ß@'oÙ¸Ù$"27§ÞX#2ß@ØX/ "7)ØÁä2Ùß@4Ù q2ØÁa/"Ù¸Û
ÞXs
0 = β
12äβ 6= 0
'aÙfæñ òuÙBä2Ùä2 "ØX ÞX#2ß@oæñaÛ3Ù ""ÙãÄHcX[]ÒWbcO×T O×£ HÞX#2ß@
2x +2y +2z = 12 3y +6z = −3 5z = −10
%¬-:
& " Ù "u¿/Á2e
:" H7âÛOæéØÁa/ÙØÁ3ÛOæç"äÙÁ2
$¤ O2 "! å BÙîB 2ä2ÙX"L@ä"27
a ij
82ä
i > j
Ù7Á ¥
ä<à "2äÙL@ä2"7â2 åÛ"!Ùa/ ")Ù7Ù,Á 2ã
â¸2 ÞX#2ß@&q 2ÙÛë2ß@272'4u ÙVÜÙ äußÖÁ3Ù ""ÙOãaêáÙä2`Û§ß@ÙÛ2Û
2Ù ""Ùe
¦hrkut&iwvur|k)t ubay2Xk
ÛÙ) å "à ÛeÙä1/ "ä2 "
z
Ý wæéÛÛ "qÛ #2 ØÁa/ÙO'"y
Ý3 \æñ/"ÛàÛâ ")Û2#2ß@2ØÁa/"ÙuáÛ å/ " ¯Ûz
'7"¯2Max
Ý wæé"ÛÛ å) 2ß@"#2 2ØÁa/"Ùu Ûà/ " àÛy
uÛz
ã(ð)Ù$27ä2ez = −10/5 = −2
'"y = 1 3 (−3 − 6z) = 1 3 (−3 + 12) = 3
Ma
x = 1 2 (12 − 2y − 2z) = 1 2 (12 − 6 + 4) = 5
ã+ sÞ qÛÙä)2 ")Ù ""ÙÖ4Ù$ "e(5, 3, −2)
ãX2'
(5, 3, −2)
2$2{Ù ""ÙBÛ2 Ù2ØÁa/"ÙâÛ¸ÞX#ßÖ/ãHÞX#2ß@@Ù Ù,"ØÁ@e(5, 3, −2)
㨧©kumªSiwvut k3t uay2Xk
å ""72 7qÛoæéBsÙ"7ÛX3Ù ØX)eaÙf /ïbÙ ßÖ 3ÞX#2ß@
%¬-:2{¸ÞX#2ß@2ØÁ¿/ "Áâä2Ù "ß@ "/ãHH$)2)ÛOæçÙ$2 âÝÖ å«Ma{¸ÞX#2ß@1äÛ
äÙLÖä"Áq% â åÛ"!Ùa/ ")uÛ
0
Ý åïbÙ"{ÛÙ)Û å&Û"!Ùa/ "
äÙß@ßÖä/æé)ÛÜÝ "3ä1¬
u/BÛ22ã4.¯, /"ØÁ'a, 7Û,ÞX#ßÖ""
2 2 2 12 0 3 6 −3 0 0 −5 10
%8í
ÙfäÙßÖß@2ä23a ß@ Û2
1
å&Û"!Ù "eL 1 ← 1 2 L 1
L 2 ← 1 3 L 2
L 3 ← − 1 5 L 3
1 1 1 6 0 1 2 −1 0 0 1 −2
%
"âÙBïZ/â ª Û2
0
XÛÛ åÛå/!Ùa '2{äÙß@ßÖQ®17a åÛ #2 ä2Ù "Ùãsð) Ù$"73eL 1 ← L 1 − L 3
L 2 ← L 2 − 2L 3
L 3
1 1 0 8 0 1 0 3 0 0 1 −2
"
L 1 ← L 1 − L 2
L 2
L 3
1 0 0 5 0 1 0 3 0 0 1 −2
ä/æéÝÛ" &e
x = 5
y = 3
z = −2
%“:
¯¬° 33±,kC²,>B#8,?46>}"33³1'+3$,?3²>}EF#)*-,?3/$µ´='->}2>}$&¶·*1'P4=34¨$&46,?´¤'+3|!3¸>}2>}$&¶·*1'P³¹."!²4¨;¥,?´='+3/$¹º»*-,4=>¼'·3|A46>),?3/$t."!²'1>&4
(x, y, z)
'·>}$µ>&´=!$&4=3ºª,?´=#8>Y3½,PEF"#)5+³+*P'+3/$
x = 5
º
y = 3
'l$
z = −2
9»¾µ±'·>}$¿Cª"!#8."!²44=´EÀ,!$µ'·3$&D³·#)4¨'(;"³+#)4ÂÁ²'·#./!'t´='µÃ}5·,?3²A4=A,?$¬Ä
./!±"3|,~$)#&"!;³1'·>}$(Å4¨'·3I>}"´¨!$)4¨"39ÆH3|EÀ,?4¨$·º5+*P*P'P3|´T±,A4Â$5+4ÂÇÈA'¹>&>&!²>·º»"3C»'+!$t*1"3/$&#)'+#(./!±!²3¸>&2>O$)¶+*P'-./!4¤;³·#&4¨Á²'1´¨'¹>
C#)C#)4¨³+$&³¹>BÉT,"ÊlºÉFŪÊ'+$ÉÀ5·Ê%5+4ÂÇÈA'¹>&>&!²>%,$&!?ËO!#8>!3't>&´=!$&4=3~'l$¿A"3²5B3k'·>}$Y>&Ì#./!'¼´='YÃ&5+,3²A46A,¥$¥ÄY$)#&"!;³BC²,#Y´=,1*P³l$)DA'
5l4¨ÇÍA'·>)>}!²>%'¹>O$YÅ4='+3½>&´=!$&4=3A!½>&2<>}$&¶·*P'9
\ÞX#2ß@@Ý& wæçX"Û2ä3Ù ÙØX@e
(5, 3, −2)
ÄHcX[]ÒWbcO×T O×£ HÞX#2ß@
x + z = −1 y − 2z = 2
%
& Ma¿)ÛÙ Ù)e
z
s2 2Û 3Oæç"ß@4Ù ØÁ2 ")/ " 2'ä|a/ØX3/ 2 Ûz
Û2 ß@"BØÁ3ä2Ù "
(x, y)
2 HØÁ(x, y, z)
2Ù ""ÙÛ,Þ#ßÖ/ã4Ù $2f ")Ù" 'a 2ßÖ ØX2Ú3ØÁÞ#ßÖ %
2âØX/ "7âÝfe
x = −1 − z y = 2 − 2z
¯æéß$
S
Û2Ù ""ÙÖ \æéß$ ",Û2 "u(x, y, z) ∈ R 3 2 "ÖØÁ x = −1 + z
y = 2 + 2z
z
sÙ/7 2Û oæéßÖsÙ )ØÁ2 ")/ 2 22 "<uãað)BÙ
S = {(−1 + z, 2 + 2z, z), z ∈ R }
\ å&!TM åß@ò2ß@ä|ÙØÁ
S = {(x, y, z) ∈ R 3 , x = −1 + z
y = 2 + 2z}
ã`Ö/ "$
z
ÛÖ}iZ2rke42 " s2) 2Û Oæç"ß@4Ù ØÁ2 "/ "2 2ã_H2/ å$ 2x
y
Ù7Û¿}iw|k|j@e "2 à/ "2 Ù7âÛu2 ßÖ2a åq/ 2 Û
z
ã) "hÞX#2ß@
%¬ì: '¬" XOæçÞq1/¥ä/ å$ h "$ ãV) "ÞX#2ß@
%
u'V" 7Þ1/¿¥?/äßÖ7
/ "$ " "$ ã)qÛOæñ/ 2Þ#ßÖ2'¯" ás2ÞÅ1Ù" "" / å/$ " "$ 22ã ßÖ '
Ûa/ "ÞX#2ß@
x 1 + 2x 4 + 3x 5 = 2
x 2 + 2x 5 = −3
x 3 − 2x 5 = 0
%¬6:
Þ@Û2/ å$ 2¯ "$ 2e
x 4
u
x 5
'7 Ù"/ "$ 2¯ "2e
x 1
'
x 2
'7
x 3
ã`Þ#ßÖâ4hâ 2ä2 "
Ù åïbÙ ßÖe
x 1 = 2 − 2x 4 − 3x 5
x 2 = −3 − 2x 5
x 3 = 2x 5
¯æç22ß$
S
Û2fÙ ÙÖ2f \æéß$ §Û2ØÁÁ(x 1 , x 2 , x 3 , x 4 , x 5 ) ∈ R 5 "ØÁ x 1 = 2 − 2x 4 − 3x 5
'
x 2 = −3 − 2x 5
x 3 = 2x 5
x 4
x 5
sÙ/7â Û Oæç"ß@4Ù ØÁ " "/ "2 ¥ ãsð)
ä2 ¿3e
S = {(2 − 2x 4 − 3x 5 , −3 − 2x 5 , 2x 5 , x 4 , x 5 ), (x 4 , x 5 ) ∈ R 2 }
.B2 ä"ä
2sÙfÛa å¸ÙÛf$a/@Ûa!<e
qÛu ß@" & wæç"ØÁ Ù ÙÌÛ¹ÞX#2ß@
2ä2ÛÁ " "ØX
x 4 = 1
x 5 = 0
Î $HàßÖòßÖØÁÙ,82äx 4 = 0
ux 5 = 1
ãQÏÄHcX[]ÒWbcO×T O×ÀÐ \Ù"Û ÙâßÖÁ2a/Á ÞX#2ß@3¿/Áe
x 1 + 2x 2 + 3x 5 = 2
x 3 + 2x 5 = −3 x 4 − 2x 5 = 0
%89
Ñ·ÒY³+C»3²>&'·>¼0ÉÀ,"ÊB0
(0, −3, 0, 1, 0)
ÓÉFŪʼ0
(−1, −5, 2, 0, 1)
ßÖ#2 3X's "Ù ""Ù4Áa ª" 3 "ÛÀL@ä2" 2âÝ@ Ù2 âØÁqÛ 3ÞX#2ß@ %¬6»u'
ßÖ"2ïZ¿1'Väæç2h?ä2ß@27á "àß@ò2ß@ÞX#ßÖ/'7ï4ØÁOæçÙ@)ä|a! "h è àÛ2¯ä2ÙÁ2ãHÞX#2ß@
s2â 2ä2 3Ù åïbÙ ß@e
x 1 = 2 − 2x 2 − 3x 5
x 3 = −3 − 2x 5
x 4 = 2x 5
%
¯æç22ß$ 3Û2âÙ Ùà
S = {(2 − 2x 2 − 3x 5 , x 2 , −3 − 2x 5 , 2x 5 , x 5 ), (x 2 , x 5 ) ∈ R 2 }
RBSkurzuiwz|k
4Ù3Ûaâ åÙ3Û3$aÛa!eaÛu2 ßÖ2 \æé"ØÁ3Ù ""ÙÛBÞX#2ß@ ä22Û27
2 " ØÁ&e
x 2 = 1
'x 5 = 0
Î $Hx 2 = 0
'x 5 = 1
ãH\Ùß@a 2 Ý \æé?2 ä"ä)ßÖ å 4Ù à "ÞX#2ß@Û \æé?ßÖ "3íã Ô`PaOÕ 1-rv gfYrqYQÖ~fYjlsy~ykqYrfv×qYQÖ~fYj+sty~ykqYroQqYm~ukx8vr
O×FÐ`×&¡ JUK¥^YbÀ¬YbÊs^d
ð) ä2)ßÖ"72a7 åïbÙ ß@ÛÞX#ßÖ3/ØÁ2 OÙÙa)$sÙ" Ûa "ä2à!2 wã)
2Û2{Û"Ma¿Ù¿/72'(Ù,Û" &ØÁOæç ""!ÛOæç,$ "2BÛ3ä2ÙX"L@ä2272-Ù,Á ":` "2
äÙLÖä"Ám&psrkut&iwkurt kut@2rk3¿â ä2u3 "!3Ù7âaÙX "2ã
JUK^YÀ¬YZÊs^á× âÈ$ "2ëÛÖäÙL@ä2"72Û¿f|z|aku}"yVnano&áä|aØÁÖ ""!&ÙëÁ " @äÙß@ßÖä2a
Ûa1/7|/!Û
0
ØÁ3 å ä22Û27/ãoâÞX#2ß@ "2" 2ä|2 "Ù3O |/$ "2Ûä2ÙX"L@ä"27ä2Ù 2sÙÛ72ä|2 Ù/ã(ð)fs2 "âpsiyVj3ÛOæçfÞX#2ß@2ä|2 "Ù3 "àä2ÙX"L@ä227ØÁOa "ÁBò3Û "!2àÙ,Á "2ã
JUK^YÀ¬YZÊs^á×»§$ "2§Û&äÙLÖä"Á3Û¿Ö|z|aku}"yVnanoqr|mXi)æé ¥22ä| "Ù'H¥ "2ÙÙ7
Ù2!fÝ%'O "ä2ÙX"L@ä2272XÛâÛ2ÙÙ7Á "ãaÞ#ßÖ3 "1/" 2Û
ä|2 Ù ÛH ")$ "2fÛ3ä2ÙX"L@ä227äÙ 2sÙÛ7â2ä| "Ù3 2Û2ã
RBSkutps}"k|jØ
"2à ÙÞX#2ß@2âÛ
%Ù7âa2ä| "Ù2ã
")ÞX#2ß@2&
' 6:
%¬-:Ù732ä| "Ù23ß /")a/2ä| "Ù23 Û2ãYHÙÙ7 2
s2äu2ß@27e% u&ìB4Ù @ÞX#ßÖ
&
'%Ö % sÙ "@ÞX#2ß@
6: '`-X'íB&ìBsÙ "@ÞX#2ß@
%¬-:uã
"ÞX#2ß@x +3y −z = 8 4z = 4
â2!7/ "2ß@27ä|2 Ùß /"àa/âä|2 Ù3 2Û¿1ãHH¿ÙâÙ7â2!7BÝ
1
u4
ãÙ¹ÒY³+C»3²>&'·>¿0»ÉT,"Ê%0
(0, 1, −3, 0, 0)
ÓÉFŪÊ0
(−1, 0, −5, 2, 1)
9 °
3-Å$)4¨'·3"$´¨'¹>H*PÚ+*P'¹>_>&´=!$&4=3²>./!'¿A,3²>H´£±'+Û'+#85l465l'C#)³·5+³·A'·3/$
Ü
!3'(C»'+#)*!$8,¥$&4=3A'¹>Y4¨3ª5l3²3/!²'·>C#)¶·>·9
[kut@r2m(k@eaBÞX#2ß@2ä| "Ù 2Û¿4ä2Ùß@4Ù Û2 "!ÛfÞs
0 = 0
Ù0 = β
12äβ 6= 0
ãa 2ß@ 'a ")$ 1Û3äÙLÖä"Á/7â2â2ä| "Ù 2Ûe
1 0 2 0 3 1 0 1 6 0 1 0 0 0 0 1 2 4 0 0 0 0 0 0 0 0 0 0 0 5
-5
H3ÞX#2ß@äÙ 2sÙÛ7e
x 1 + 2x 3 + 3x 5 = 1
x 2 + 6x 3 + x 5 = 0 x 4 + 2x 5 = 4 0 = 0 0 = 5
-%
âÛÙä3ä|2 Ù Û2ã
`Pa)Ý Pstj+uvx+sty~rm|{}u~yr-rvgfqYQÖkfYjlstyky~qÞoQqYm~ukx8v
JUK^YÀ¬YZÊs^ÅeÁÛaÖÞX#2ß@) "1 ä|2 ÙÙ ä|2 Ù) 2Û1'Á 2á/ å/$ "ØÁ(äÙ 2sÙÛ27
äÙ "Ù2àÙî" sOæçÞaàÛ)ÙÙ7Û¿2}iZ2r|k|juãH2à/ å$ 2ØÁOä2Ù sÙÛ7/äÙ "Ù2
Ùîf oÞf&BÙâÙ7âÛ2}iw|k|juã
RBSkutps}"k&eÛ "ÞX#2ß@
%89 ' 2à/ "$ 2 "$ 2hÙ7
x 2
x 5
' 2/ å$ "2h "22Ù7
x 1
'
x 3
x 4
ã
RBSkurzuiwz|kqeÁÛa/ âÞX#ßÖ
-%u'ÁØÁ2 "2ÙÁh "/ å$ "2á "$ 2HGØÁ " 2Ù7h "2á/ "$ 2á "22¤Gkß
H2ÖÙ Ù&ÛOæçDÞ#ßÖ,2ä| "Ù, 2Û¿ B Ù27 Û,ßÖ#2 BßÖß@2Ûå/ã(àXâ fÞX#2ß@
äÙßÖsÙ {¸ "!fÛDÞXs
0 = β
1äβ 6= 0
'häÙß@ßÖB " ÞX#2ß@2
%%&
-%u'" OæéÅaÖÛ
Ù ""Ù2ãHàÙ§e
3æé OæÞ̸a/&Û / "$ 2q "$ 2'¥ "ÖÞX#2ß@B¸ä2ß@27Ù ""ÙÅØÁà ÛßÖ"#
ßÖß@2Û"//'äÙß@ßÖ)Ûâ ")Þ#ßÖ
%¬ì»uã
3æé hÞ̸ÙÌ 2 q/ "$ "q "$ 22'" áÞ̸" MaÖÛÙ ""ÙÖe "q/ "$ 2q "$ 2
s227 2Û Oæç"ß@4Ù @ØÁ2 "/ " 2 " 2& 2/ å$ "2 "2æé? "ß@27&ÅïbÙäu"Ù]Û
/ å$ 2à "$ 2'ä2Ùß@ß@)4Ù "2ÞX#ßÖ
% '
%¬6»
%89:uã
¯"ØÁÖ 23Ù ""ÙÛoæéëÞX#2ß@@2ä| "Ùu 2Û Ù7qÛßÖ#2 &ßÖß@2Ûå/'s" L
sÙ 2ÙÛ â@ÞX#ßÖ "2" hÛ ïbÙ ß@2 ÞX#2ß@âä|2 Ù 2Û¿¯2ØÁ¿/ "Á2ã`/7
ÛßÖÙ7 2 hØÁ)äæç2Ù¬ÜÙ h4Ù$ "/'XÙ8ÞÙäÙß@ßÖÁä2 åqs2àïZ â ØX2 "ØÁ2ß@ 22ã `
ß@ÙÛ" "2)Ûa/ "&2äuÙ,/7æñ/4 " ßÖuÙXÛ3Ûá/+âÙ ÛaOã_\æç2/ å7Û
"ßÖuÙXÛ3Û~Ûf¿ÙÛ|á3/ã
ã·ÒY³+C»3²>&'·>¼0²´¨'¹>Y;¥,?#)4=,Å´¨'¹>´¨4=Å#)'·>Y>&3/$
x 3
'l$
x 5
9ä'·>Y;¥,#&46,?Å´='·>´=4¨³·'·>Y>&3/$Y´¨'¹>Y,?!$)#&'¹>¼0
x 1
º
x 2
'l$
x 4
9
+ oÞf ""2 ßÖ"# àÛ ÙÛ 3fÞX#2ß@3 "1 "ÞX#2ß@2ä2ÛÙ Ù7âa
ïbÙ ä2ßÖÁ 2Ù "Û@ åß /"# & å ä22ÖØÁáÙ2ã`.áÈ u/ä|'` "Bß@ÙÛ&ÛÖ 2Ù ""Ù
ÙsÙ f wæé1/Á!ÛOæçò ÙqÝïZ3!2 "
Ù "B2äu"ÙÅ-ãñ6: ã`.á "ä2ÙßÖsÙ &Û2]a e
Ûa/q "B 2ß@"# '`ÙÈßÖu "@ÞX#2ß@ ÙqïbÙ ßÖ2ä|2 "Ù2/ã)/q "BÛ"#ßÖ/'Ù]ß@ "@ÞX#2ß@
ÙïbÙ ß@2ä|2 "Ù2 Û¿ã
ÄHcX[]ÒWbcO×TëO×&¡¢ì2ÙÛ 3 ÞX#2ß@/7
3x + 3y + 3z = 18
−x + 3y + 7z = −10
x + 3y + 4z = 6
--:
.¯ä2 ÙÙÛoæñ$sÙ ÛB ÞX#2ß@ÙïbÙ ßÖÞßq$4Ù "ØÁ&e
3 3 3 18
−1 3 7 −10 1 3 4 6
-í
¦hrkut&iwvurkàpaVj8kXk}¿Ör|j8yV}mX\iwyVns
Hä2ÙX"L@ä2272]Ýf!7ä|ÛëÞX#2ß@@2æés2 " & "&¿Ù3ÛÈÞ#ßÖ/ã¥ð)ëä2Ùß@ß@2ä&a 3 "
ïbÙ ß@2 ,
1
,ß ""7 " ßÖ#2 ""!)a1/3
ãL 1 ← 1 3 L 1
L 2
L 3
1 1 1 6
−1 3 7 −10 1 3 4 6
-
âïbÙ"ØÁ 3Ùâ22! Ý
1
'aÙ,ïZ¿âa/ ª Û20
Ù 3Ù ä2ØÁ` u27Ý@ ""ß@"2 à "2ß@"# "ä2ÙÁ3ÛÛ2fÛ2 "#2 2â2ØÁa/"Ù¥uã
L 1
L 2 ← L 2 + L 1
L 3 ← L 3 − L 1
1 1 1 6 0 4 8 −4 0 2 3 0
-ì:
ð)¸ 1/" " q qÙÏÞX#2ß@@äÙß@4ÙÛÛ2¸Û2 "# 2ØÁa/"Ù
ÛÙä&Û2Û§Û2
#2 2 ""!2HÛ|$ "1/ÛäÙL@ä2"7¥ ã²`¿ÙÛä2ÙÞX#ßÖh2 "¯äÙL@ä2"7
4
ãð)ä2Ùß@ß@2ä a/ ") ïbÙ ß@2 ,1
2BÛ¿7 "Û2"#ßÖ "!)a4
eL 1
L 2 ← 1 4 L 2
L 3
1 1 1 6 0 1 2 −1 0 2 3 0
-
" Ù ïZ¿a ²" f5]Ù ,Ù/
ä{ØÁâ u27fÝë2 "ßÖ2 & "ÈÛ"#ßÖ,"äÙÁ{Û{ å
Ù"#ßÖ2ØÁa/"ÙH
L 1
L 2
L 3 ← L 3 − 2L 2
1 1 1 6 0 1 2 −1 0 0 −1 2
-6:
äÙßÖß@¿Ùe
L 1
L 2
L 3 ← −L 3
1 1 1 6 0 1 2 −1 0 0 1 −2
-9
©kumªSiwvut kàpaVj8k
Û~a/3Û3 ßÖÙ72/eaÙ,ß@ ÞX#2ß@3ÙïbÙ ßÖ)ä|2 Ù Û¿ã
H)$2ß /"7a7ÛïZ" /a ª" âÛ2
0
ÛÛ2Ù ä2ØXo uX"27Ûàä)ä1Ý ""ß@"2 å Ù""#2ß@3"äÙÁÛ2)Û2{ ßÖ#2 ØX/Ùu) "ÖÛ"#2ß@qä2ÙÁÛ åÖ 2ß@"#ØX/ÙHuãÙ ä2 å'ÙBä2Ùß@ßÖä2 ïZ" a/ ª Û2
0
BÛ2âÛ]XkurunaiwkurqÙ1ãL 1 ← L 1 − L 3
L 2 ← L 2 − 2L 3
L 3
1 1 0 8 0 1 0 3 0 0 1 −2
-
"ÙfïZa ª" ),Ú2 ÙÖBÛ2âÛBÛ2#2ß@3Ù3e
L 1 ← L 1 − L 2
L 2
L 3
1 0 0 5 0 1 0 3 0 0 1 −2
í5
ð)B2 Ù à fÞ#ßÖ
%¬ì»uã+ OÞf&ÛÙä33Ù ÙØÁ
(5, 3, −2)
ãÄHcX[]ÒWbcíO×TëO×T ) \æé?2ß@ ") 2ä2Û71'( "àÙOæéu|27¥ÜßÖ"hÁ "ãQH$Û \æé?2ß@ "
ØÁHÛ3ßÖÙ7 2 äÙß@ßÖÁäÙ7X2 å 2Ù ""ÙØÁaÛ{ÙÙß$4 BÙÁ wã
à4ÙÙâØÁOæéÙB2 " " 2ÙÛ ÞX#2ß@/73e
2x 3 − 2x 4 + 8x 5 = −6
x 1 + 2x 2 + x 3 + 5x 5 = −1
−2x 1 − 4x 2 − x 3 − 8x 5 = −1
í%
ð),äÙß@ßÖä2a/ â ßÖu ÙïbÙ ß@ÞXß$sÙ "ØÁ&e
0 0 2 −2 8 −6
1 2 1 0 5 −1
−2 −4 −1 0 −8 −1
H)¿ÙhÛÞX#2ß@
"âä2ÙX"L@ä2272 aàÝq!7ä|hà!7 4Ý
0
ãð)4àÛÙä)ah "2 á "2ß@"# "!sÙ 2 "ß@" " 2ß@"# ä2ÙÁàÛÛ2#2ß@u Ù""#2ß@á2ØÁa/"Ù2ãÙ 2ÙÛ
ä Ù$ #2ß@'ÙBä|a! " ßÖ#2 uâ å&Û2#2ß@ "!e
L 1 ↔ L 2
L 2 ↔ L 1
L 3
1 2 1 0 5 −1
0 0 2 −2 8 −6
−2 −4 −1 0 −8 −1
ð)fïZa ²" )Û
0
Ù "¿Ù3eL 1
L 2
L 3 ← L 3 + 2L 1
1 2 1 0 5 −1 0 0 2 −2 8 −6 0 0 1 0 2 −3
í7-:
u " 2ß@"# 3ä2Ù "Ùãs*{Ù{Û å&ä|aä2 eaäÞX#ßÖä2Ùß@ß@2ä ä2Ù Ù3Û5ãsð),
s2àaàß@ Ûa1/Á!)Û5q ä2)äÙ "Ùãð) 1/ " "ÛÙä)Û 2äußÖÁà à ÙÏÞX#2ß@
Ù$2ÁB2B ß 7 " ßÖ#2 ""!u 2Ûf 2ß@"# 2äÙ "Ù2ÛfÞ#ßÖ
í7-»uãQH3¿ÙÛ
äÞX#ßÖ2â2!7 `Ý@-ãaðBä2ÙßÖß@2äa " 2Û 32!7/ `Ýf%e
L 1
L 2 ← 1 2 L 2
L 3
1 2 1 0 5 −1 0 0 1 −1 4 −3 0 0 1 0 2 −3
"ÙBßÖuÛ2Ú2 ÙÙ "¿Ù
L 1
L 2
L 3 ← L 3 − L 2
1 2 1 0 5 −1 0 0 1 −1 4 −3 0 0 0 1 −2 0
HÞX#ßÖ2à "Ù áÙïbÙ ß@2ä| "Ù1ä)Û2%ä2ÙßÖß@Ùãð)32 ßÖâ å3 ßÖ#2 âa/ã
`3Û2#2ß@ä2Ù"Ýß@ Û% Û2Û2Ù/2'Á ä2ÙßÖß@2Q®27a "Û2 " áÙ1ã
L 1
L 2 + L 3
L 3
1 2 1 0 5 −1 0 0 1 0 2 −3 0 0 0 1 −2 0
L 1 − L 2
L 2
L 3
1 2 0 0 3 2 0 0 1 0 2 −3 0 0 0 1 −2 0
ð)B2 ¿3BÞX#2ß@32ä| "Ù 2Û¿3e
x 1 + 2x 2 + 3x 5 = 2
x 3 + 2x 5 = −3 x 4 − 2x 5 = 0
íí
\æç2?/äßÖ7 )ÞX#2ß@
%89 㤯æéß$ "
S
Û2Ù ÙeS = {(2 − 2x 2 − 3x 5 , x 2 , −3 − 2x 5 , 2x 5 , x 5 ), (x 2 , x 5 ) ∈ R 2 }
`Pa8ï æçxlrfm¸{Oukyr-rvgfprstukrè¥s(oQgfqYQÖkfYjlstyky~qYfÞoQqYm~ukx8vfévÖ~qYsto xlf
ST¬ÊsÒ¯Êsd1YÀ¬YbÊs^ðO×}¡pñ,itpayVr k2maku}sju~8juvut k)}inoir|kpakumX¯j8k)t ku\Zrkàpr)mXnokjumXiky|paurZiwyVnju}"ut kuna
wirk|jj8yVmÁj3mXnokáx1yVr t k|z|aku}"yVnano|kk|zXk|j
1
z|yVt&t kàpsiyVj2ST¬cÃHÕsc(×Hð)BïZ/ 2ä 2äq â Ùß$ ÛOæç"äÙÁÛBÞX#2ß@2ãàÙ
H p
å Ù "&e
H p
eoæéßÖsÙ ØÁ HÞX#2ß@ "2" )Ý
p
"ä2ÙÁ4ò ïbÙ ß@)a ÛOæçÙ4 V"Ù "ßÖÁ" 2,BÞX#2ß@2ä| "Ù 2Ûâ1äÛ
1
äÙß@ßÖ¿Ù2㯠ÙÙhØÁ
H p
2áX /"4Ù ¯Ù
p ∈ N ∗ã` 2Û H 1
h å2â@ "2äu
sÙ á Ù
H 1
" o LÛ ÙXä22Û2 ä2Ùß@ß@Ûa å 2Û
H p ⇒ H p +1
ä2ÛÙ¥ ãs*,Ù7 Ù
H p ⇒ H p +1
ã
à4ÙÙ
H p
X /"ã¤\ÙÛ2 ÙBÞX#2ß@ "2"
(S)
Ýp + 1
"ä2ÙÁ2ãSTVcX[ÅYbcÁTBÂ_adeo å 2ß@"# ä2Ù ÙäÙß@4Ù "ØÁßÖÁ)Û2äÙLÖä"ÁÁ "'Oäæç2ÝÛ" q
)|$ 1fÛä2ÙX"L@ä227â2Û "qïbÙ ß@&e
0 ∗ .. ∗ ∗ .. .. .. .. ..
0 ∗ .. ∗ ∗
"ÞXß$sÙ
∗
áä1"!TMaàØÁâ "äÙL@ä2"7h áä2uä22ØÁ "äÙØÁ ãðäÙÛ#2 Ù & fÞX#2ß@{ "1/"(S 0 )
Ù$X 2 "ßÖ7@ å§ 2ß@"#2 fä2Ù Ù/ã1\BÞX#2ß@{ "2" p
ä2ÙX24ÛÙä'a à7ÞX4Ù/#Û) 2ä 2ä'aò ïbÙ ßÖ2Þ#ßÖØX/ "7
(S 00 )
ä|2 Ù&12ä&Û%qäÙßÖß@ÙãO.¯¸ 8ÜÙ|/ÁäÙ "ÙÛ
0
/¸ÞX#2ß@(S 00 )
'4Ù{Ù$27)Þ#ßÖ2ä| "Ù82äÛ2%3ä2Ùß@ß@ÙØÁ`2â2ØÁ/ 27âÝ
(S)
ãJcXÃ`Ä`YbÇX[]c½Â_adBeá "23äÙLÖä"Áq " ßÖ#2 äÙ "ÙÖÖÙ7qa/ÙX "2ã)/äÖä22'
ØÁ¿3Ý2ä|! âÛ2B "!2'ÙBs23 ß@2 âÝ,Þ#ßÖ3ÛfÞX4@e
a 0 11 ∗ .. ∗ ∗ a 0 21 ∗ .. ∗ ∗ .. .. .. .. ∗ a 0 n 1 ∗ .. .. ∗
1ä
a 0 11 6= 0
ã(.¯BÛX7 å 2ß@"# "!aa 0 11
'aÙfÙ$27BÞX#2ß@ÛBÞXs@e
1 ∗ .. ∗ ∗ a 0 21 ∗ .. ∗ ∗ .. .. .. .. ∗ a 0 n1 ∗ .. .. ∗
"4Ù Ù
i ∈ {2, ..., n}
'Ù,Ù ¿a 0 i 1
ïbÙ" "& ""!
1
Ý@ "& ""!i
ã4ð)BÙ$27â Ù ,ÞX#2ß@Û3 "qïbÙ ß@&e
1 ∗ .. ∗ ∗ 0
..
0
∗ .. ∗ ∗ .. .. .. ..
∗ .. ∗ ∗
í
àÙ¿
(S 0 )
"ÖÙÞX#ßÖ 2ä2Û ä2¿ÏÛ2'äæç2Ý{Û @ "ÖÞX#2ß@Ù$2Á½Ý{a ÛÌÞX#2ß@í
2, ß 7 " 2ß@"# "! å 2ß@"#2 3ä2Ù Ù/ã
(S 0 )
2BÞX#2ß@Ýp
"ä2ÙÁ2ãs¥/7ÞX4Ù/#Û3 2ä 2ä'sÙf4âÛÙä3 ïbÙ ß@2 a ÛOæçÙ4 /"Ù2 "ßÖ7| 2,
Þ#ßÖâ2ä| "Ù12äâÛ2â%äÙß@ßÖàÙ/2ãX.¯ïZ"7á "2áÙs2 /Ù¯äÙ 2sÙÛ72 ÞX#2ß@
í
u'ÙfÙ$27âBÞX#2ß@ä|2 Ù1äÛ3%äÙßÖß@)¿ÙØXH2âØX/ "7Ý
(S)
ãò å,eÛa/Ùà "ä22'
H p+1
àX "/ãÙä'a à 2ä2 ä2/'
H p
2àX sÙ hÙ
p ∈ N ∗ã
[kut@r2m(ke "q Û) å ÙsÙÙÖïbÙ ¿ßÖuÙXÛ4Ù h ïbÙ ßÖ hÞX#2ß@ ""1
ØÁ "ä2ÙØÁ2&ÞX#2ß@à2ä| "Ù)eäÙß@ßÖÛa¥ å 2'Ù /ßÖ# h Ù$ #2ß@àqïZ/Ûß@
ÙàïbÙ ßÖ2ä| "ÙÞX#2ß@
(S 0 )
ØÁHä2ÙÁÛ)ß@Ù2'XàÙf ß@#2) " Ù$ "#2ß@Û ß@(S 0 )
ÙïbÙ ß@àä|2 Ù& Ù$ #2ß@àÛàß@ àÙáïbÙ ß@2ä|2 "Ù2&ÞX#ßÖäÙ727 ä2ÙÁ3Û3ßÖÙØÁ(S 0 )
'auä/ãmXnokjumikqy|paur\iwyVnju}"ut kunaÏirk|j2
STVcXÃHÕscs×qæñ/ # " ÙsÙÙ¸ 2ä2ÛÁ'Ù«4ßÖu &Oæç"ß@4Ù qØX2 áÞX#2ß@ÖÙïbÙ ßÖ
ä|2 Ù 1äÖÛ2
1
ä2Ùß@ß@2¿Ù2ãH+ ¯2 "Ù ïZä" "qÛ " ïbÙ ß@2 ëÈÞ#ßÖ@2ä| "Ù 2Û¿
" ( LÛïZ a ª" Û
0
Û2Û Û2 " ¿Ù2'"à/Û2Û wæé1/7Û2 "2¿Ù2'ä'(ä2ÙßÖß@)Ûaâ 2ßÖ 2Û3 å2äu"Ù¸-ã
ã
ô õùYõfúhùõe÷Zf÷zÁöÏ÷þBøH÷çö^÷÷Zf÷ø øYXø2õùXqþføùZÅöõfþ øOÿøOù ÷zþ^?¥÷|ú
a+b 1-rv gfóÖksgstûtYykfntrrs-x+qóüýukyþr-rvgfójlx+y~qYtxlo ftÿoQty~û m¸{Oukyþr-rvgf
jlxlykqYtxoQf
JUK^YÀ¬YZÊs^ëe7ÞX#2ß@ayVt yXvunok
Ùfj2njj8k|z|yVnst kut@2rk¥a "2¯äÙL@ä2"7ÛÖ2äÙÛ@ßÖß$
Ù7âÙÁ "ã
¥/ â?2ß@ "/'aÛa
%u' ÞX#2ß@Û3!7/ä|32ÙßÖÙ!#2/'ß /"àa/â 2 2ã
JUK^YÀ¬YZÊs^§eVÙ¿
(S)
&ÞX#2ß@ ""1 ãHju~8juvut khayVt yXvunokVj|j8y8zuiw/ÞX#2ß@(S)
2¯ "hÞX#2ß@ØÁ` ")ß@ò2ß@ 2ß@"2 ß@2ßq$ ØÁ
(S)
ß /"àÙîf 2äÙÛ,ßÖß$ âÁ wã RBSkutps}"keHH3ÞX#2ß@ÙßÖÙ!#2ÙXä23fÞX#2ß@3x + 6y = −3
−2x + y = 12
3x + 6y = 0
−2x + y = 0
JUK^YÀ¬YZÊs^
ÒT¬Ê(Ò¯Ê4d2YbÀ¬YbÊs^àÙ
(S)
@ÞX#2ß@Ûn
2ØÁa/"ÙÝp
"äÙÁ2h(S 0 )
"ÞX#ßÖâÙß@Ù/!#/ÙXä2ãàÙ, ïbÙ ßÖ
(S 0 )
2{¸ÞX#2ß@qä|2 Ù
Ù,2ä| "Ùq Û¿?)ØÁOæçÙ¸ "ßÖ
2¯ "!
0 = 0
'Ù@Ù$27á@ÞX#2ß@Ûr
ØÁa/ÙãXð@ÛßÖu
Ù" ¯ä2Ù áÛ@ä2ÙÛÖßÖ ØÁ
ä7"
r
2¯"Û2s2Û7Ûh åß "#2 hÛÙ7ÙqÙs2 /ãð) wæés2 " rnX3ÛqÞX#ßÖ ""1(S)
ãRBSkurzuiwz|k
2sÙÛa åBÙÖÛ $qÛÖa!<@eØÁ2 h2q "@ !¸Û2ÞX#2ß@2 O
u'
%%u'
%¬-»u'
%“:u'
% '
%19
í%uã
[kut@r2m(k|jØ
,ÞX#ßÖ3 "2" ) ÞX#2ß@ÙßÖÙ!#2ØÁ` "o2)/ÙXä2Ù7âßÖòßÖ !ã ÆH3 A±,?!$&#)'½$&'·#&*P'·>·º_3±4=*PCª"#}$)'./!'+´B>}2>}$&¶·*1'ø³·58D'·´¨"33³³¹."!²4¨;¥,?´='+3/$Ü
(S 0 )
'l$k3'5+3/$&'·3²,?3/$wCª,>wA'´¨43²'·>
0 = 0
5l"*1C»#&$&'Y´='Y*PÚ+*P'3"*Å#)'YA±³·./!²,¥$)4¨"3²>·9/¾'Y3*Å#)'%>·±,?CC»'+´=´¨'Y´='#8,?3 (A'
(S)
9/ÆH3PEÀ,?4¨$·º"3PC»'+!$_*PÚ·*1'Y*P"3"$)#&'·#H./!'º"C²,?#)*P4
´¨'¹>Y>}2>}$&¶·*1'¹>%3't5l"*PCª"#}$8,?3/$Cª,>YA'(´=4"3'·>
0 = 0
º4=´'lÛ46>O$)'(!3!346."!²'¼>&2>O$)¶+*P'(³·58D²'+´=33³B#)³·A!4¨$³·./!4=;¥,?´='+3/$
Ü
(S 0 )
9
·ä'·>#),3 ">%>&3/$·ºA,3²>%´£±"#)A#)'º ºº²º ºº 1'+$9
(S)
s2&ò Û"Mahä2Ùß@ß@@ ÖÙß$ ÛÖÙ Ù]ÛOæéä2ÙÁ2 "2¥)Ûa/Oæç"ß@sÙ )ØÁ2 `ÞX#2ß@32ä| "ÙØÁ/ 27âÝ
(S)
ã /!)2¯"ïb2 2 Ù!7 Ùß$ ÛOæéØÁa/Ù'XØÁOæé&Ùß$ Ûoæéä2ÙX2¯ÛÞX#ßÖer ≤ min(n, p)
ãr = n
'Ûa/ÞX#2ß@2ä|2 "Ù Û¿à2ØÁ¿/ "ÁàÝ(S)
' (4a/Þ1Ù" àÛ) "!2hÛÞXs
0 = β
ãHHÞ#ßÖÛß@ÛÙä,ß@Ùà3Ù ""Ùoãr = p
'" oOæçÞÛOæéä2ÙÁ2â "$ 2ã¤H3ÞX#2ß@@ÛÙäf "Ù ""ÙOãr = n = p
ÞX#ßÖ3ä2 Û !@ßÖ²"ßÖ Íu'7 "ÞX#ßÖ3âÛÛø\ ßÖ /ã+ H&äu2ß@27 Ù ""ÙOãr < p
'" 4ÞßÖÙ"à"äÙÁ "$ ã¤àXo "ÞX#2ß@âä2ÙßÖaV"$ "'Á O&ÛÙä" M Û3Ù ""Ù2ã " /!ÛOæé,Þ#ßÖOæç"ÛØÁa'a2B!2 \'Hä3ÞX#2ß@âä2Ùß@a/"$ )ÙfÙoãa)` Pstj+uvx+sty~rm|{}u~yr-rvgfÖksgstûtYykf
ñyVÏZiwyVnj,eHÙ27Ä
= (x 1 , x 2 , ..., x p )
uÆ= (y 1 , y 2 , ...., y p )
Ûp
Û@ 2 "2'oλ
ë 2 wãð)¸Ù/q 24ä¿ßÖÁ
λ
ÄÌuÄ+
ÆÌ "p
Ïλ
Ä= (λx 1 , λx 2 , ...., λx p )
)Ä+
Æ= (x 1 + y 1 , x 2 + y 2 , ..., x p + y p )
ã\Ù"Û Ù)§ÞX#2ß@&Ùß@Ù!#
(S 0 )
u)ÙÙS 0
\æéß$ qÛ2)Ù ""Ù2ã¯æéß$ "
S 0
Ù" Ù "u2ïbÙÛaß@27| 2e
:
S 0
ÙX"Û
,ou1'
(0, 0, ..., 0)
Ù¬ÜÙ Ù ""Ù¤Î
$¤OÄ
= (x 1 , x 2 , ..., x p ) ∈ S 0
'a Ù sÙ àÙ 2
λ
'λ
Ä∈ S 0Î
ä<àOÄ
= (x 1 , x 2 , ..., x p ) ∈ S 0
Æ
= (y 1 , y 2 , ...., y p ) ∈ S 0
'( Ù hÄ
+
Æ∈ S 0
ã
RBSkurzuiwz|k
Û"ä2/ÙàÛ åÙ/3Û3$aÛ3a!)e Ù 2 Ù "2
$¤u
ä<uã
`@ Ù "u
$¤à@äÙ2ØÁ2ä2qßÖß@2Ûå/qe
ST¬ÊsÒ¯Êsd1YÀ¬YbÊs^ð O×}¡ g3kunj8kut@2}"kXk|jâj8yV}mX\iwyVnj)mXnfju~8juvut kâayVt yXvunok3k|juj8yVihw7}o
{(0, 0, ..., 0)}
Hj8yViinánaiZ
STVcXÃHÕscs×
(0, 0, ...0)
2ÖÙ¬ÜÙ @Ù ""ÙOãá{ 2'á2æç" âB,Ù Ù¹ÙDÁ "į'h "Ù fe ÛOæç@a 1'OÛoæñ #2 å Ù "u
$H 'OsÙ Ù3 2
λ
'λ
Ĺ٠٠ΠÛOæé a 1'HäÙßÖß@qÄ 2 Ù,Á \' wæç22ß$ "3Û2p
Ï Û "ïbÙ ß@λ
ÄÈ1äλ
2 â"MaÎ
ÛÙä3 "Þ#ßÖ@3" Ma
Û3Ù ""Ù2ã Q
$}$&'·3/$&4=3½0">&4²!²3>}2>}$&¶+*P'Y'¹>O$_A'¼¾#),*1'·#·º¥4=´ª,¼'+Û,5$)'+*P'+3/$_!²3'>}"´¨!$&4=39,?46>¤!3>&2>O$)¶+*P'YCª'·!$,¬;"4=#¤'lÛ,5l$&'+*P'·3"$H!²3'
>}"´¨!$&4=3->&,3²>QÚl$)#&'A'B¾#),*P'+#¹9¥ÆH3-':'l$¹º3±4¨*PC»#&$&'./!'·´²>}2>}$&¶·*1'5+*PC²,?$&4=Å´¨'A3/$H´¨'¿#),3 ¼'·>}$H³",´,?!P3"*Å#)'A±4¨3²5+33<!'¹>
,1'lÛ,5l$&'+*P'·3"$Y!3²'t>}"´¨!$)4¨"39
"!#-C#)!;"'+#½ÉFŻʺH4=´¿>&!"!P$wA'½#&'·*-,?#8."!²'+#P."!²'>}4
a i1 x 1 + a i2 x 2 + ... + a ip x p = 0
,´¨"#)>1C»!#P$)!$#)³+'·´
λ
º3e, 0
a i1 (λx 1 ) + a i2 (λx 2 ) + ... + a ip (λx p ) = λ(a i1 x 1 + a i2 x 2 + ... + a ip x p ) = λ × 0 = 0
9 #µ'(*1Ú·*P'ºC»!#YC²#&"!;'·#¼ÉT5·Êlº4¨´>}!"!P$
A'B#&'·*P,#)./!'·#_./!'¼>}4
a i1 x 1 + a i2 x 2 + ... + a ip x p = 0
'l$
a i1 y 1 + a i2 y 2 + ... + a ip y p = 0
º<,?´=#8>
a i1 (x 1 + y 1 ) + a i2 (x 2 + y 2 ) + ... + a ip (x p + y p ) = (a i1 x 1 + a i2 x 2 + ... + a ip x p ) + (a i1 y 1 + a i2 y 2 + ... + a ip y p ) = 0 + 0 = 0
9
%$¥° 3A4Â$¿./!±!3't>&´=!$&4=3'&
= (x 1 , x 2 , ..., x p )
'·>}$)(+*,(-(."/0/1>}4´='·>
x i
3'(>&3/$YC²,>$)!²>Y3<!´6>·9
$s1ä2Ù{ "e(4Ù Û" ØÁOæç¸22ßq$ "2 TMa " Ù "2
$H
ä<u'sÙ{Û¿âØXoæé juÏ72}"kpr
z|yVt@2insi"j8yVnÌ}inoirkuãÌÙÏ22ß$ Û
R p ØÁÙÅXÛ u&$ " a qäÙß$aÙÈ ""1 æñ4 " "Bj8yVm7juk|jpXz|kk|zuyVr iwku}4Û
R pãH2 Ù 2
'
$Hu
ä2æç "ß@27hÛÙä2ÖÛ"7ØX
}kunj8kut@2}"k3Xk|j)j8yV}mZiwyVnjqmXn,ju~¬juvut kayVt yXvunok
p
inoz|yVnanam(k|jk|juhmXn{j8yVm7juk|jpXz|k3k|zuyVr iwku}`XkR pã
& 2'sÛaOæéßÖsÙ ØX2 Þ#ßÖä|2 Ù&ØÁ/ 273Ý
(S 0 )
's "qÙß$ ÛOæç"äÙÁ2 "$ 22! Ý
p − r
'Ùîr
& /!§Û(S 0 )
ã¼H2Ù ""ÙqÛÌÞX#ßÖæé? ßÖÁÛÙäÅïbÙä"Ù Ûp − r
/ å$ "2'aØÁs227 2Û oæéßÖsÙ ØÁ2 "2/ " ã(+ïbÙ ß@2 "ßÖÁ2'aÙ{s2Û ØÁ \æéß$ "S 0
Û2àÙ ""ÙÛ
(S 0 )
p − r
Û2! 2hÛ "$s2 Vã`Ù"Ù/!âÛ-Û! Û ""$s2 : 2 2ß@ å/ä2)/2äÙÛ2ß@2 a/ à åÙÙÛÛßÖÙ/'7uÙÛ ØÁS 0
Ùaä
2äÙ 2 HÛ
R p Xkqit kunjuiwyVn
d = p − r
ãð)B2 ØXoæéB H2ä2&3 äu $2fa ä2 "#2 e
d = 0
r = p
'S 0
3ä2Ù7"27âØÁ3 "sÙ"7
{(0, 0, ..., 0)}
ã
d = 1
r = p − 1
'S 0
2Û Ù¿a7âa
(0, 0, ..., 0)
ã
d = 2
r = p − 2
'S 0
2B "ä2Ù72a7
(0, 0, ..., 0)
ã
d = 3
r = p − 3
'S 0
2B2aäÝ& Ù"Û"ß@2"Ùä2Ù7a7
(0, 0, ..., 0)
ã
d > 3
'S 0
2,aä½ÛfßÖòßÖ)ÞXsß /"àÛkÛ"ß@2"ÙY "! Ûã
Ù
d ≥ 1
'` ïZ3ØÁ@ wæç22ß$ "S 0
Û2Ù ""Ù3ÛOæçÈÞX#2ß@ÖÙßÖÙ!#2@¿3Û"ß@2"Ù
d
¿!TMaØÁ3Oæç"ß@4Ù ØÁ2 "Ù ""ÙÛäÞX#2ß@s22æç ßÖ ä2Ùß@ß@wäÙß$a"Ù "2" âÛ
d
Ù ""ÙÛ$:/'O3ØÁOæç" ¥Oæé3a/4Ù$ "Ûoæé? "ß@2 Ù2 2Ù ""ÙÛ«ÞX#2ß@ä2Ùß@ßÖ äÙß$a"Ù "1/" Û3ßÖÙ"Ûd
Ù Ùã¤\3ÙXä1$ å )2 2ä2Bä2ÙÛ,ßÖ ã.B2ß@ "eaä2ÙÛ2 Ù "Þ#ßÖ3Ùß@Ù!#Ùä"3BÞX#2ß@
íí:e
x 1 + 2x 2 + 3x 5 = 0
x 3 + 2x 5 = 0 x 4 − 2x 5 = 0
¯æç22ß$ 3Û2âÙ ÙàÛä3ÞX#2ß@e
S = {(−2x 2 − 3x 5 , x 2 , −2x 5 , 2x 5 , x 5 ), (x 2 , x 5 ) ∈ R 2 }
+äw'
p = 5
'r = 3
ÛÙäd = 5 − 3 = 2
ãHð)§4? "ß@2 Oæç"ß@4Ù ØX2 " qÙ ""Ù,2§ïbÙäuÙÛ2 Ù ""Ùd2 = (−2, 1, 0, 0, 0) d 5 = (−3, 0, −2, 2, 1) Ù$2Á) s2äu2ß@27sÙ
x 2 = 1
'x 5 = 0
'usÙ
x 2 = 0
'x 5 = 1
uãa.¯B"41'a \æéß$ Û2âÙ Ùàs2æéä2 " 3Ùâ åqïbÙ ßÖ&eS 0 = {x 2 (−2, 1, 0, 0, 0) + x 5 (−3, 0, −2, 2, 1), (x 2 , x 5 ) ∈ R 2 } = {x 2
d
2 + x 5
d
5 , (x 2 , x 5 ) ∈ R 2 }
êáÙ2 2ÚBä2ÙÛBßÖ ØX
S 0
2æç"72 #!2Ùß@ "ØÁßÖ7ä2Ùß@ßÖ, åOã
.B2 ä"ä easÙ Ùâ 2ÞX#2ß@2 "1 ØXÙâ8Ùä2Ù"Û 2'(Û2 ßÖ2 å&ÛßÖÙ
d
Û@ wæç22ßq$ "ÛÙ ""ÙÛ]Þ#ßÖÖÙß@Ù!#2Ö/ÙXä2ãHÙ ØÁ
d ≥ 1
'H? ßÖ \æéß$ @Û2 Ù ""Ùh ïbÙäu"Ù Ûd
Ù Ù¯ " 2ØX) wæçÛ2/ å$ "2 "$ ¯/1
uà 2à/ 2h/ å$ "2 "$ 2à/ "70
ãð)]XëØXoæéÈÞX#2ß@ÖÙßÖÙ!#21/¿)Ù¬ÜÙ 3/Èß@Ù") åÙ ""Ù
(0, 0, ..., 0)
ãO.¯È /ä|'BÞX#2ß@ÙfÙß@Ù!#Oæé@a/ïbÙ ä2ßÖ7ÛÙ ""Ù2ãðBÛ¿ "Ù ØÁOæç" O"äÙßÖ/$ "/ã
RBSkutps}"kea "ÞX#ßÖ/7
x + y = 5
x + y = 7
í7ì:
â"äÙß@a/$ "/ã
îÙ "ÙßÖ"7a7Ù ØÁ)ØÁaÛ "ÞX#ßÖ2àä2Ùß@a/"$ 'Á wæç22ß$ "Û)àÙ Ù2
2 ")Û3ßÖ"# 2ä"Ý& wæç22ßq$ "ÛÙ ""ÙàÛBÞX#2ß@ÙßÖÙ!#23/ÙXä2ã
àÙ¿
(S)
{ÞX#2ß@q "2" ãHàXÙ(S 0 )
"3Þ#ßÖÙß@Ù!#2qÙXä2ã_àÙ¿S
S 0
wæç22ßq$ "3Û
Ù ""ÙàÛ
(S)
Û(S 0 )
'a 24äßÖ71ãsÙ ÙÄ= (x 1 , ..., x p ) ∈ R p'ÙBÙ
Ä
+ S 0 = {
Ä+
Æ,
Æ∈ S 0 } = {
Ä0 ∈ R p , ∃Æ ∈ S 0 ,
Ä 0 =Ä +
Æ }
+
Æ}
\æéß$ "Û2â "Û 2 "Û3 åïbÙ ßÖÄ
+
ÆÈ82ä3Æ∈ S 0
ã
ST¬ÊsÒ¯Êsd1YÀ¬YbÊs^ð O×T32(i¯}"k)ju~8juvut k}inoirk
(S)
7t kuh}¿&j8yV}mZiwyVn{¶ }"yVrjS =
¶+ S 0
STVcXÃHÕscs×àÙ¿
(S)
"ÞX#ßÖ!2 "ØÁ
-: ãQàÙÄ«)Ù ""ÙÖÛ
(S)
ã*,Ù7 ÙS ⊂
Ä+ S 0
a
ÛÙ$ "3ä2 ÙOãa*BÙ7 ÙâÙâÛOæé$sÙ Û
S ⊂
Ä+ S 0
ãàÙÄ
0 = (x 0 1 , x 0 2 , ..., x 0 p ) ∈ Sã54oÊ4YÀ3Æ
=
Ä0 −įã
àÙ
i ∈ {1, ..., n}
ãð,ea i 1 y 1 + a i 2 y 2 + ... + a ip y p = a i 1 (x 0 1 − x 1 ) + a i 2 (x 0 2 − x 2 ) + ... + a ip (x 0 p − x p )
= (a i 1 x 0 1 + a i 2 x 0 2 + ... + a ip x 0 p ) − (a i 1 x 1 + a i 2 x 2 + ... + a ip x p )
ð) àÄ
0
ÄÈÙ7âÙ ""ÙÛ(S )
'(ÛÙäa i 1 x 0 1 + a i 2 x 0 2 + ... + a ip x 0 p = b i = a i 1 x 1 + a i 2 x 2 + ... + a ip x p
Ùä
a i 1 y 1 + a i 2 y 2 + ... + a ip y p = 0
ã%\ÙßÖß@qä2ä2¥2 ¥sÙ Ùi ∈ {1, ..., n}
'sƽ٠""Ù{Û(S 0 )
ã(ð) Ä0 =Ä +
ÆáãaÙä3Ä 0 ∈Ä + S 0
+ S 0
'aÛÙä
S ⊂
Ä+ S 0
ã
*,Ù7 ÙâßÖ"7a7àÄ
+ S 0 ⊂ S
ãHàÙ¿Æ∈ S 0
Ä
0 =Ä +
ÆáãaÙ Ù i ∈ {1, ..., n}
'
a i 1 x 0 1 + a i 2 x 0 2 + ... + a ip x 0 p = a i 1 (x 1 + y 1 ) + a i 2 (x 2 + y 2 ) + ... + a ip (x p + y p )
= (a i 1 x 1 + a i 2 x 2 + ... + a ip x p ) + (a i 1 y 1 + a i 2 y 2 + ... + a ip y p )
= b i + 0
= b i
ÙäÄ
0 ∈ Sã(ÙäÄ + S 0 ⊂ S
'âa ÛÙ$ ä2 ÙO'
S =
Ä+ S 0
ã
`, ÙsÙ¿"Ùë 2ä2Û7Ö 27Û "fßÖ#2 @/7,eá}¿Bj8yV}mX\iwyVnëXunour}"k mXn]ju~¬juvut k
}inoirkk|juhXyVnano|kàpr}¿j8yVt&t k3mnokj8yV}mXZiwyVnÖpr \iwzumX}iwvurkXkz|k)ju~8juvut k&kuhXk}¿j8yV}mXZiwyVnfXunour}"k
mju~¬juvut kayVt yXvunokVj|j8y8zuiw2
mXnokinánaiXk)j8yV}mZiwyVnj2
STVcXÃHÕscs×%àOfÞ#ßÖ "2" 2ä2Ùß@a/"$ ' "Ù 2'XÛOæé #2 åq ÙsÙ¿Ù 2ä2Û27qíãñ-' wæç2
2ß$
S
ÛhÙ ""Ù22 $Ü2äuÙ12äâ wæç22ßq$ "ÛhÙ ""ÙÛ@ÞX#2ß@ÙßÖÙ!#2/ÙXä2ã qæé #2â å ÙsÙ¿"Ùfíã%'S
&ÛÙäÙ,2 `2 2ß@271'Ù¿3" Mã á2Ùß@ "ØÁßÖ71' \æé!7S =
Ä+ S 0
Û åq ÙsÙ¿Ùíãñ-q2æç"7 #Û åqßÖ#2 /7&e
(Ä,2Ù ""Ù&Û
(S)
'X "Ù ¯ wæç22ßq$ "ÛhÙ ""Ù¯Û(S)
h wæç"ßÖ! á å) "/ÙÛä Ä Û wæç22ß$ "&ÛÙ ""ÙÛëÞX#2ß@@ÙßÖÙ!#2@ÙXä2ã84OY wæç22ßq$ "&ÛÙ ""Ù)Û(S)
2Ù@X"Û'7" 3ÛÙä åß@ò2ß@ïbÙ ßÖ:ØX \æéß$ "Û2Ù ÙÛÖÞX#2ß@ÙßÖÙ!#2/ÙXä2ã\æç2
ÛÙäÙ§4Ù"7
r = p
u'sÙ¿Û Ùr = p − 1
u'sÙ¿§ år = p − 2
u'sÙ§2a/ä2àÝ Ù"`ÛßÖÙ
r = p−3
'8Ù¿2aä2(ÛßÖòßÖ¯ÞXshßÖ"HÛ ! ÛhÛ"ß@2"ÙOã QQRBSkutps}"k\Ù"Û Ù )ÞX#2ß@
(S)
e
2x 3 − 2x 4 + 8x 5 = −6
x 1 + 2x 2 + x 3 + 5x 5 = −1
−2x 1 − 4x 2 − x 3 − 8x 5 = −1
í
H3ÞX#2ß@ÙßÖÙ!#23/ÙXä2
(S 0 )
e
2x 3 − 2x 4 + 8x 5 = 0
x 1 + 2x 2 + x 3 + 5x 5 = 0
−2x 1 − 4x 2 − x 3 − 8x 5 = 0
í76:
.¯B 2Ù /7
(S 0 )
Ùf Ù3ØÁ \æéß$S 0
ÛÙ ""ÙàÛ
(S 0 )
eS 0 = {(−2x 2 − 3x 5 , x 2 , −2x 5 , 2x 5 , x 5 ), (x 2 , x 5 ) ∈ R 2 }
"2'
(2, 0, −3, 0, 0)
2âÙ ÙÛ(S)
ãH¯æç22ß$ 3Û2âÙ ÙàÛ(S)
âÛÙäS = (2, 0, −3, 0, 0) + S 0 =
(2 − 2x 2 − 3x 5 , x 2 , −3 − 2x 5 , 2x 5 , x 5 ), (x 2 , x 5 ) ∈ R 2
ð)]2 §Å2ä2ÙÛ½2ß@2 ØÁ
S
2æç"7 #!2ÙßÖu ØÁ2ß@27äÙß@ßÖ@Å "O'äæç2Ý{Û aäqÝ&ÛBÛßÖÙ2ã[kut@r2m(k|j@e
åÖÙ ""Ù
(2, 0, 3, 0, 0)
qa/)BÞ24'OßÖ"Ûa/ä2 "ÞX#2ß@2" HÞ,ÖqÙ Ù uÛ27㯠èqØÁÛÖ 2ÙÛ "ÖÞX#2ß@"" w'4" á2 Ù "q /"ÛÖÛ 2ÙÛ "ÖÞX#2ß@ÙßÖÙ!#2Ùä"
ØÁÖ \æñ1/7|/!ØÁ "3ä2ÙÛ§ßÖß$ 3ÙVÜÙ âX w'(ÛÙäß@Ù"Ûä1 ä2 âÝ
ïZ <hâÛOæç "2 å Ù4Ù"Ùfíã-ã
&9 4²´='¿>&2<>}$&¶·*P'3±'·>}$_C²,>HD*P:"¶+3'"º?,?´=#8>
(0, 0, ..., 0)
3±'·>}$_C²,">H>}"´¨!$&4=39¾3²>&4=A³+#)3²>QC²,#H'lÛ'·*1C²´¨'Y!3~>}2>}$&¶+*P'Y´=4¨3²³·,?4=#)'
(S)
3"3~D²*P:"¶+3'"º<5+*PC²,?$&4=Å´¨'"º<'l$¿A'¼#8,?3
r = p − 1
9äH±'+3²>&'+*Å´='¼A'·>%>}"´¨!$&4=3²>YA!>&2<>}$&¶·*P'µD"*P:¶·3'¼,>)>}5l4=³B'·>}$Y,´¨"#)>
!3'tA#)4¨$&'¼C²,">&>),?3/$%C²,?#
(0, 0, ..., 0)
'+$Y´T±'+3²>&'+*Å´='tA'·>Y>&´=!$&4=3ª>YA'
(S)
!²3'tA#)4¨$&'¼3'(C²,">&>),?3/$%C²,>%C²,#
(0, 0, ..., 0)
9
ÛÛä2 wæç22ß$ "Û2Ù Ù2ã( â2ß@ "/'
(0, 1, −3, 0, 0)
)Ù ""Ù,a/ ä2 "# Û(S)
ãð),ÛÙä
S = (0, 1, −3, 0, 0) + S 0 =
(−2x 2 − 3x 5 , 1 + x 2 , −3 − 2x 5 , 2x 5 , x 5 ), (x 2 , x 5 ) ∈ R 2
\æç2Ö @ß /"# Û@Û2ä " Ö \æéß$ @Û2qÙ ""ÙØÁ ä2 " ÛÙÖ "a2'HÙ
/ "$ "/ã
RBSkurzuiwz|keaßÖÙÁ 2B Ù7aÛÙ$ ""ä2 ""ÙØXoæéÙ{&$"
(2 − 2x 2 − 3x 5 , x 2 , −3 − 2x 5 , 2x 5 , x 5 ), (x 2 , x 5 ) ∈ R 2
=
(−2x 2 − 3x 5 , 1 + x 2 , −3 − 2x 5 , 2x 5 , x 5 ), (x 2 , x 5 ) ∈ R 2
RBSkurzuiwz|ke_\Ù"Û Ù ÞX#2ß@
(S)
/7e
2x 3 − 2x 4 + 8x 5 = −6
x 1 + 2x 2 + x 3 + 5x 5 = −1
−2x 1 − 4x 2 − x 3 − 8x 5 = −1
−x 1 − 2x 2 − 3x 5 = 4
í9
*,ÙÁ ØX "ÞX#ßÖÙßÖÙ!#2)Ùä"Ý
(S)
Ý "2ßÖòßÖÙ ""ÙhØÁ ")ÞX#ßÖ(37)
'ßÖØÁ(S)
Oæé,aÛ Ù ""Ù2ã;)Ù]$"Û"" ,e`ä2ÙßÖß@27 wæé 3Û ä2 "!2VÏ ä2Ù ä2u? ä"ä2/'( ØÁïZ )Û3ä2 "ä "'aÝ&a " Û3 \æé?ßÖ ") 2ä2ÛÁ2ã
< õÿøOù ÷øý>=÷ú7? ½¼ø àùY÷ø
ð)]äÙÛ#2 ÖÛä2@äÙÛ2ÞX#2ß@2 ""1 2ä2Ùß@4Ù |/ÁÛ ß@# 2'`ä2Ùß@ß@& "2
Þ#ßÖ/7'aÛOæç"äÙÁ2
x
uy
'Û3a ß@# 2a
ub
ã3x + 6y = a
−2x + y = b
3x − a + 6y = 0
−2x + y − b = 0
3x − a + 6y = b
−2x + y − a = 0
Õwa+b @^(oQgvoQfYrtu rfYstykmgfgn~oQfpukykxlPu~fgfYyv
HÙ ØÁOæé&ÞX#2ß@ "1/" ¯ä2ÙßÖsÙ áÛa/ ß@# 2ØÁOæéq2ä2ÙÛ&ßÖß$ '¬ "áß@ ÙïbÙ ßÖ
ä|2 Ù&&4Ùa/ÛÛÀLÖä 2ãHð)§ ÙXä2#Û&ä2ÙßÖß@æé ¥OæÞ§1/¿)aÛa ß@# 2ãOïbÙ"
ØÁ ")ÞX#ßÖ2ß@"ÙïbÙ ßÖä|2 Ù 2Û/'" 4Þf&Û2Bä23e
%1 qä1/ e ÖÞX#ßÖ äÙßÖsÙ @aqÛ "!Û]ÞXs
0 = f (
a ß@# 2)
Ùîf (
ß@#)
Û! "Ù@ØÁÛ4ÛÛa ßÖ#u 2
ä2ä1/ ÙÛ¯Ù|ß@ß@27 Ù ØX
r = n
uãä@ä22'H " 2ä2ÖÛ&a ß@# 2ä|a/! 2ÌeH2æç" Þ«Û23 "!)Û§ÞX4
0 = α
1äα 6= 0
'OÞ#ßÖOæéÛ3Ù Ù2'"ÙO'ÙB "ß@ "Ù Ù2fïbÙäuÙÛâ ß@# 2ã
-/Û ä1)eXs "ÞX#2ß@)ä2ÙßÖsÙ
k ≥ 1
""!2Û ÞXs0 = f 1 (
ß@#)
'0 = f 2 (
a ßÖ#u 2)
'¿ãã¿ã'0 = f k (
ß@#)
'XÙîf i (
a ß@# 2)
Û2!â "ÙØÁsÛ4ÛÛa /ßÖ#u 2ãHâïZ/ØÁ qÞX#2ß@äÙßÖsÙ qÛ2) "!Û¸Þs0 6= 0
s23/ "Ù Û4Û &Û å/ " Û2a ß@# 2ãOð)sÙ ¿"ae1Á 2O/ " `Û ß@# Ù7`2 "HØX
f i (
ß@#) = 0
sÙ OÙi ∈ {1, ..., k}
'Ùf ß@23/È%1 ä1ã_à"ÙO' ÞX#2ß@OæéaÛÙ Ùã
RBSkurzuiwz|k3eÁ2 ßÖ2 ¯ 2¯Ù ""Ù¯ÛÖÞ#ßÖ
(S)
/72ïbÙä"Ù&Û2¯/ " áÛ ß@#22
a
'b
'c
'd
ã
2x 3 − 2x 4 + 8x 5 = a
x 1 + 2x 2 + x 3 + 5x 5 = b
−2x 1 − 4x 2 − x 3 − 8x 5 = c
−x 1 − 2x 2 − 3x 5 = d
í
àÙ Ùe
Üa "¯ä1 ä2 ¥ àÞX#2ß@2á2ØÁ/ 27áÞ#ßÖÙïbÙ ß@àä|2 Ùâ 2Û
x 1 + 2x 2 + 3x 5 = −b − c
x 3 + 2x 5 = 2b + c x 4 − 2x 5 = −a/2 + 2b + c
0 = −b − c + d
+ OÞfÛÙäÛ2Bä23ea
d 6= b + c
'a "ÞX#ßÖOæé@Û3Ù Ù2ã¤àd = b + c
"Û2 "# 32ØÁaV"ÙÙ¬ÜÙ /ïZ¿)u \æéß$ Û2Ù ""Ù
S = {([−b − c − 2x 2 + 3x 5 ], x 2 , [2b + c − 2x 5 ], [−a/2 + 2b + c − 2x 5 ], x 5 ), (x 2 , x 5 ) ∈ R 2 }
Õwa)` @^(oQgvoQfYrtuhko fgx+f%ogfgkoQf
`Ba/Ù§2 "Û2 "ä1/ãH.¯ë"42'H4Ù ß@ &ÈÞX#ßÖ@ÙïbÙ ß@&2ä| "Ù2@ Û'
Ù]ÛÙa ïbÙ"3ïZ @Û2&Ùs2 /ÙÛ]ÞXs
L k ← λL k
'
$¤
L k ← 1 λ L k
ÙÅ 2 / å7
æ6
L k ← λL k + L i
'
$Oæ=
L k ← L k + λ 1 L i
'Ùî
λ
@½ä2ÙX"L@ä"27@Û½ 2ß@"2 qßÖß$ /ãáð) 2'λ = 0
'ÞX#2ß@Ù$2Á½2ÅïZ/"/Á wæçÙs2 /Ù
ÙÌ wæçÙ4 /"Ù
æ63Oæé&Ì!22 / aq2ØÁ/ 27&/
Þ#ßÖ&å
Î
ØÁa/Û«¸Ù4 /"Ù
$¤
$Oæ=u'o " 2oæéÙ7ä2 2ß@27a/Û2'HuØÁaÛ§Ù«
\æéßÖ 2"Ù
ïZ/ØÁOæçÙ«4 " )ÛÙ2 «22'O ÞX#2ß@&ØÁOæéÙëÙ$27Oæç232§!2 a
ØX/ "7â,ÞX#ßÖ"" wã
` #2! ÛOæéÙ å/7ea}"yVrj22máyVnkumXxirkâmXnoky|paur\iwyVnBmÖ\~|pak
L k ← λL k
yVm2iwkunBi8i"j8kur
2maku}¿2m(k z|ay8j8kpr
λ
)ku2m(kλ
k|jumnok k·Spsr|k|j|juiwyVn]Xpakuns7naâXk)prt vuZr|k|j| ài}xmX)i"juZinXm(kur&}"k zVjλ = 0
m§zVjλ 6= 0
) "qä2
λ 6= 0
'oÙ§s2ïZ" q wæçÙs2 /ÙBÙ ãH) ä1λ = 0
'4Ù§s2)a2'oß "Ù " "ïbÙ ß /"Ù
λ = 0
Ùâs2 ßÖuÛ34Ù X 3 " 2Ù ""ÙOã ì2ßÖ ØÁÖes$"2{Û""!2 \æéÙs2 /ÙL k ← L k + λL i
'aÙVÜÙ 4 ß@"3ØÁaÛ
k 6= i
'sÛq \æéÙsu /ÙL k ← λL k + L i
'"72 ÛØÁa/Û
k 6= i
uλ = 0
ãÙ X2 ÛOæé8Ù" à ÙBÛ3ä1/Ý&Û"!2 Û#â 2 2ß@"#2 2Û3 " 2Ù ""ÙO'Ùfs2e
•
.áä|! âÛ2B "!ªMaBÛïZ 3 ª )BÙØÁH3ä2Ùß@ 2ÛBaÛ3a ß@# 2•
î3aÞX2ßÖ/"ØÁ2ß@27â3 ßÖ2 /,ä2Ùîf "¿Ù/1
ã•
qæçqß "#2 3!2 'sq" " åÖß@ÙXÛÛ¸ÙÛßÖ"# 3 !"Û'aßÖ"â wæéÛa2 ªM,Û3Û"!2 à ßÖÙÛ3ä1/âsÙ"$ "2ã