@~7ABe?t3X35O@ A NO.2 MWLs:#F|$N%F!<%^ Cj>]E*%Y%/%H%k6u4V!"%Y%/%H%k6u4V$NItJ,6u4V!#0l<!
FHN)@<B?tD>@~$b!"  
1t;t ∈ R
1$b!"!VF1$87A!W$r$7$F$$$k!#$3$N$h$&$J#2$D$rF1;~$K07$&$N$K$O!"@.J,$r8+$k$N$G$O$J$/!"OB$H!"%9%+%i!<G\$H$$$&F;6q$N$_$rMQ$$$F5-=R$9$k$3$H$,Bg;v$K$J$k!#Nc$($P!"@.J,$,$9$Y$F 0 $N%Y%/%H%k (0 %Y%/%H%k) $O v + v = v $N2r$H8+$k$3$H$b$G$$k!#$=$b$=$b!"@.J,$r07$C$F$P$+$j$$$k$N$G$O%Y%/%H%k$r8&5f$9$k0UL#$,$J$$!#
W0 =Dj5A 2.1 (652J=q5.1 $*$h$SDjM}1.1 ). V $, K >e$N%Y%/%H%k6u4V$G$"$k$H$O!"$D$.$N@-<A$rK~$?$7$F$$$k$H$-$K$$$&!#O (1i;;$NB8:_) V $K$O!"OB$H!"%9%+%i!<G\ (K $N85$K$h$kDj?tG\)$,Dj5A$5$l$F$$$k!#
I (V $OOB$K4X$7$F2CK!72$G$"$k!#)(1) ∀x, ∀y, ∀z ∈ V $K$?$$$7!"(x + y) + z = x + (y + z).
(2) ∃0 ∈ V $,$"$C$F!"∀x ∈ V $K$?$$$7!"x+0 = x, 0+x = 0$,$J$j$?$D!#(3) ∀x ∈ V $KBP$7$F!"∃y ∈ V $,B8:_$7$F!"x+y = 0, y+x =0 $,@.$jN)$D!#(4) ∀x, y ∈ V $K$?$$$7$F x + y = y + x $,@.$jN)$D.
II (%9%+%i!<G\$N:nMQ)(5) ∀c1 , c2 ∈ K∀x ∈ V (c1 c2 )x = c1 .(c2 .x).
(6) ∀x ∈ V 1.x = x.
III (OB$H%9%+%i!<G\$N6(D4@-)(7) ∀c1 , c2 ∈ K∀x ∈ V (c1 + c2 )x = c1 x + c2 x(8) ∀c ∈ K, ∀x, y ∈ V c(x + y) = cx + cy.
L?Bj 2.1. %Y%/%H%k6u4V V $K$?$$$7$F!"(1) V $N85 v $G!" v + v = v $rK~$?$9$b$N$,$?$@$R$H$DB8:_$9$k!#$3$l$, V $N%<%m85(%<%m%Y%/%H%k) $G$"$k!#(2) V $N3F85 x $KBP$7$F!" 0.x = 0 $,@.$jN)$D!#(3) V $N85 x $KBP$7$F!" x + y = 0 $r$_$?$9y $O$?$@$R$H$D$G$"$k!#$3$l$r x $N5U%Y%/%H%k$H$h$V!#
Dj5A 2.2. %Y%/%H%k6u4V V $NItJ,=89g$G!"(F1$8OB$H%9%+%i!<G\$K4X$7$F) %Y%/%H%k6u4V$G$"$k$h$&$J$b$N$r!"V $NItJ,%Y%/%H%k6u4V$H8F$V!#>e=R$N W0 $b R2 $NItJ,%Y%/%H%k6u4V$G$"$k!#

Dj5A 2.3. V $N%Y%/%H%k v1 , v2 , v3 , . . . , vt $,0l<!=>B0$G$"$k$H$O!"$"$k(c1 , c2 , . . . ct ) 6= (0, 0, . . . , 0) $,B8:_$7$F!"

c1 v1 + c2 v2 + . . . ct vt = 0$rK~$?$9$H$-$K$$$&!#v1 , v2 , . . . , vt $,0l<!=>B0$G$J$$>l9g$K$O!"0l<!
FHN)$G$"$k$H8F$P$l$k!#
MW$9$k$K!"0l<!=>B0$G$"$k$H$O!"M?$($i$l$?%Y%/%H%k$N4V$K4X78<0$,B8:_$9$k$3$H$G$"$k!#0l<!FHN)$+$=$&$G$J$$ (0l<!=>B0) $+!"$OOB$H%9%+%i!<G\$N$_$rMQ$$$F5=R$5$l$F$*$j!"@.J,$NCM$K$D$$$F$OD>@\$O8@5Z$7$F$$$J$$!#$$$+$K$b@~7ABe?tE*$J35
G0$G$"$k!#1

2@~7ABe?t3X35O@ A NO.2 MWLs

Nc 2.1. (@~7A=>B0$J%Y%/%H%k$NNc)(1) R4 $N85$r   134
2
4
6

  v1 = 3 , v2 = 5 , v3 =  8 
4
6
10$GDj$a$k$H!"v1 , v2 , v3 $O0l<!=>B0$G$"$k!#v1 + v2 = v3 $G$"$k$+$i$G$"$k!#(2) R3 $N85$r   
117




v1 = 2 , v2 = 1 , v3 = 83
1
9$GDj$a$k$H!"v1 , v2 , v3 $O0l<!=>B0$G$"$k!#v1 + 6v2 = v3 $G$"
$k$+$i$G$"$k!#(3) R2 $N85$r   
1
51
v1 =, v2 =, v3 =2
10
1$GDj$a$k$H!"v1 , v2 , v3 $O0l<!=>B0$G$"$k!#5v1 = v2 $G$"$k$+$i$G$"$k!#>e$N$h$&$K!"6qBNE*$J?t%Y%/%H%k$,0l<!=>B0$G$"$k$+H]$+$rD4$Y$k$H$-$K$O!"@.
J,$K8@5Z$9$kI,MW$,$"$k!#v1 , v2 , v3 $,0l<!=>B0$G$"$k$H$$$&;v$O!":G8e$N%Y%/%H%k
v3 $, v1 , v2 %9%+%i!<G\$NOB(@~7A7k9g) $rMQ$$$F=q$1$k$H$$$&$3$H$K$+$J$j6a$$$,!":G8e$NNc$N$h$&$KNc30$b@8$8$k!#$=$3$GDj5A$G$O:G=i$+$i v1 , v2 , . . . vt $K4X$7$FBP>N$J7A$G=R$Y$F$"$k$N$G$"$k!#>e$N (2) $G8@$($P!"1.v1 + 6.v2 − 1.v3 = 0 $H$$$&D4;R$G$"$k!#Nc 2.2.

  15v1 =, v2 =2
1$H$*$/$H!"v1 , v2 $O0l<!FHN)$G$"$k!#<B:]!"c1 v1 + c2 v2 = 0$H$9$k$H!"

c1 + 5c2 = 0, 2c1 + c2 = 0$G!"$3$l$r2r$/$H c1 = 0, c2 = 0 $rF@$k$+$i$G$"$k!#@~7AFHN)@-!"@~7A=>B0@-$rH=Dj$9$k>l9g$K$O$&$($N$h$&$K0l<!J}Dx<0$K5"Ce$5$;$k!#2<$NLdBj$b;2>H$N$3$H!#"(%l%]!<%HLdBj
LdBj 2.1.

   141
v1 = 2 , v2 = 5 , v3 = 1
3
6
1$H$*$/$H!"v1 , v2 , v3 $O0l<!FHN)$@$m$&$+!"$=$l$H$b=>B0$@$m$&$+!#M}
M3$r5s$2$FEz$($J$5$$!#