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K“R C
K
K K K E2
‚ ` EˆE E pE, `q
~
0E~ 0
‚.KˆE E p~u, ~vq P E2pλ, µq P K2
1.~u “~u
pλ`µq.~u “λ.~u `µ.~u
λ.p~u `~vq “ λ.~u `λ.~v
λ.pµ.~uq “ pλµq.~u
E u, v K
~
0E
nPN˚KnK
`, .
‚ px1, x2,...,xnq ` py1, y2,...,ynq “ px1`y1, x2`y2,...,xn`ynq
‚λ.px1, x2,...,xnq “ pλx1, λx2, . . . , λxnq
R2R3
R
Knn“1
R R C C
C R C R
C C
FpX, Eq “ EX
XKEK`.
"pf`gqpxq “ fpxq ` gpxq
pλ.fqpxq “ λfpxq
FpR,Rq “ RRR R R
~
0RR“f:"RÑR
0ÞÑ 0
CFpR,Cq “ CR
X“N RN
CNR´
C´RrXsCrXsR
C

EKλ, µ PK~u, ~v PE
λ.~u “~ 0Eô pλ“0~u “~ 0Eq
pλ´µq.~u “λ.~u ´µ.~u
λ.p~u ´~vq “ λ.~u ´λ.~v
1q2q3q
ð0.~u “~ 0Eλ.~ 0E“~ 0E
~u “1.~u “ p1`0q.~u “1.~u `0.u “~u `0.~u
0.~u “~u ´~u “~ 0E
λ.~ 0E“λ.p~ 0E`~ 0Eq “ λ.~ 0E`λ.~ 0E
λ.~ 0E“λ.~ 0E´λ.~ 0E“~ 0E
ñλ.~u “~ 0Eλ“0λ‰0
~u “1
λ.pλ.~uq “ 1 λ.~
0E“~ 0E
λ“0~u “~ 0E
EKuPEp~u1, ~u2,...,~upq
p ~u1, ~u2,...,~upλ1, λ2,...,λpK
u“
n
ÿ
i“1
λi~ui“λ1~u1`λ2~u2` ¨¨¨ ` λp~up
~u p~u1, ~u2,...,~upq
C R 1 i
RrXsn1, X, X2,...,Xn
KnrXsn`1
Kn~u “ px1, x2,...,xnq
~e1“ p1,0,...,0q, ~e2“ p0,1,0,...,0q, . . . ,~en“ p0,0,...,0,1q
~u “
n
ÿ
i“1
xi~ei
Kn
δij “"1i“j
0
KnnVect ei“ pδ1i, δ1i, ...δniqiPJ1, nK
FpR,Rq
f:xÞÑ sh x f1:xÞÑ exf2:xÞÑ e´x
f f1f2FpR,Rq
@xPR,sh x“ex´e´x
2
f“1
2f1´1 2f2
f“a.y1`b.y2pa, bq P K2

R2~u “ p3,2q~u1“ p1,´1q~u2“ p1,2q
R3~u “ p1,1,1q
~u1“ p1,1,0q~u2“ p0,1,1q
RrXsP“X P1“X´1
P2“X`1
2
R C 2`i1`i1´i
FpR,Rq
f:xÞÑ sinpx`1qf1:xÞÑ sin x f2:xÞÑ cos x
f f1f2FpR,Rq
R3
~u “ p1,4,´1q
~e1“ p1,2,0q, ~e2“ p1,0,1qet ~e3“ p´1,2,´2q
~u “~e1`~e2`~e3~u “2~e1´~e2`0~e3
n ~v1, ~v2, . . . , ~vnp~u1, ~u2,...,~upq
~u ~v1, ~v2, . . . , ~vn
~u1, ~u2,...,~up
n“2p“3
~v1~v2~u1, ~u2~u3
~v ~v1~v2
~v1“α1~u1`α2~u2`α3~u3~v2“β1~u1`β2~u2`β3~u3
~u “λ~v1`µ~v2
“λα1~u1`λα2~u2`λα3~u3`µβ1~v1`µβ2~u2`µβ3~u3
~u “ pλα1`µβ1q~u1` pλα2`µβ2q~u2` pλα3`µβ3q~u3
~u ~u1, ~u2~u3
pE, `, .qF E F
EpF, `, .q
EKF E p1q,p2q p3q p4q
F E
#piq~
0EPF
piiq @~u, ~v PF, @λ, µ PK, λ.~u `µ.~v PF
$
’
&
’
%
piq~ 0EPF
piiq @~u, ~v PF, ~u `~v PF
piiiq @λPK,@~u PF, λ.~u PF
#piq~
0EPF
piiq @λPK,@~u, ~v PF, λ.~u `~v PF

p1q ñ p2q ñ p3q ñ p1q
p1q ñ p2qF E
pF, `q pE, `q ~
0EPF
~u, ~v PF λ, µ PK. λ.~u µ.~v PF`
λ.~u `µ.~v PF
p2q ñ p3q
p3q ñ p1q p3q pF, `q pE, `q
~
0EPF
~u PFp´1q.~u “ ´~u PF
~u PF ~u `~v PF F `
pF, `q pcq.pF, `, .q
E
FĂE E F
p2q p3qFĂE E
~
0E
F F
~
0E
~u
~v
~u “λ~u `µ~v
F
EK
K
p2q p~v1, ~v2, . . . , ~vpqp
F E
p
ÿ
i“1
λi~viλiPKF
p
EKt~ 0EuE E
EK~u ‰~ 0EE
K.~u “ tλ.~u, λ PKu
E ~u
~u K~u
~
0E“0.~u PK.~u ~v1“α1.~u, ~v2“α2.~u PK~u λ, µ PK
λ.~v1`µ.~v2“ pλα1`µα2q.~u PK.~u
K.~u E
KnrXs “ tPPKrXs,deg PďnuKrXs
0PKnrXsP, Q PKnrXsλ, µ PK
degpλP `µQq ď maxpdegpλP q,degpµQqq ď maxpdeg P, deg Qq ď n
λP `µQ PKnrXsKnrXsKrXs

KE
E
pFiqiPIE F “č
iPI
Fi
@iPI, ~ 0EPFi~
0EPF
λ, µ PK~u, ~v PF i I λ.~u `µ.~v PFiFiE
i I λ.~u `µ.~v PF
F E
F G E F XG E
F
G
FXG
F1“Rp1,0qF2“Rp0,1qR2
p1,0q
loomoon
PF1YF2
` p0,1q
loomoon
PF1YF2
“ p1,1q P{ F1YF2
F1YF2R2
p0,1q
p1,0q
p1,1q P{ F1YF2
F2
F1
F1F2KE F1F2
F1`F2“ t~u1`~u2, ~u1PF1, ~u2PF2u
F1`F2
F2
F1
KE E
F1F2E
~
0EPF1~ 0EPF2~ 0E`~ 0E“~ 0EPF1`F2