PDF - 225.3 ko

E F Kf:EÑF f
E F
@~u, ~v PE, @λ, µ PKfpλ~u `µ~vq “ λf p~uq ` µf p~vq
LKpE, F qE F LpE, F q
f
f@λPR,@~u, ~v PE
"fpλ~uq “ λfp~uq
fp~u `~vq “ fp~uq ` fp~vq
fp~ 0Eq “ ~ 0F
@λ1, λ2,...,λnPK,@~u1, ~u2, . . . , ~unPE
fp
n
ÿ
i“1
λi~uiq “
n
ÿ
k“1
λifp~uiq
E E E
E
LKpE, Eq “ LKpEq “ LpEq
E F E F E F
E F E „F
E E
KE
f:"EÑF
~u ÞÑ ~ 0F
αPKf:"EÑE
~u ÞÑ α~u
α
IdE:"EÑE
~u ÞÑ ~u
R R
f:"RÑR
xÞÑ 2xg:"RÑR xÞÑ sin x
K KrXs
Φ : "KrXs Ñ KrXs
PÞÑ P1

@x, y PR,@λ, µ PR
fpλx `µyq “ 2pλx `µyq “ λp2xq ` µp2yq “ λfpxq ` µf pyq
gpπ
2`π 2q “ sinpπ 2`π 2q “ sin π“0
gpπ
2q ` gpπ 2q “ sin π 2`sin π 2“2‰gpπ 2`π 2q
aqf:"RÑR
xÞÑ 2x2bqg:"RÑR
xÞÑ 4x´3
cqh:"RÑR
xÞÑ |x|dqi:"R2ÑR2
px, yq ÞÑ py, xq
fqj:"R2ÑR
px, yq ÞÑ xy eqk:"R2ÑR3
px, yq ÞÑ p2x, x `y, 2yq
LKpE, F qKE F ~u ÞÑ ~ 0F
`.
F“KfPLpE, KqEK
Φ : "R2ÑR
px, yq ÞÑ xR2
Ψ : "KrXs Ñ K
PÞÑ Pp1qKrXs
f:EÑF g :FÑG λ, µ PK~u, ~v PE
pg˝fqpλ~u `µ~vq “ grfpλ~u `µ~vqs “ grλf p~uq ` µfp~vqs
“λgrfp~uqs ` µgrfp~vqs
pg˝fqpλ~u `µ~vq “ λpg˝fqp~uq ` µpg˝fqp~vq
E E
f:EÑF f´1:FÑE
f:EÑF λ, µ PK~u1, ~v1P
f´1pλ~u1`µ~v1q “ λf ´1p~u1q ` µf ´1p~v1q
~u, ~v PE ~u1“fp~uq~v1“fp~vqf
f´1pλ~u1`µ~v1q “ f´1pλf p~uq ` µf p~vqq “ f´1rfpλ~u `µ~vqs
f´1˝f“IdE
f´1pλ~u1`µ~v1q “ λ~u `µ~v “λf ´1p~u1q ` µf ´1p~v1q
GLpEq pGLpEq,˝q
E IdE
~u ÞÑ α~u α ‰0
GLpEq

fPLpE, F q
f
Kerf“f´1pt~ 0Fuq “ t~u PE{fp~uq “ ~ 0Fu
Imf“fpEq “ t~v PF{ D~u PE, ~v “fp~uqu
fPLpE, F qfKerf“EImf“ t~ 0Fu
fPLpEqf“IdEKerf“ t~ 0EuImf“E
fPLpE, F qKerfImf E F
tpx, y, zq P R3{x`2y`z“0uR3
Φ : px, y, zq ÞÑ x`2y`z
E, F KfPLpE, F q
fKerf“ t~ 0Eu
fImf“F
ñfKer f
~u PKer f fp~uq “ fp~ 0q~u “~ 0 Ker f“ t~ 0u
ðKer f“ t~ 0uf ~u ~v PE f p~uq “ fp~vqfp~uq ´ fp~vq “ 0“
fp~u ´~vq~u ´~v PKer f ~u ´~v “0~u “~v f
Φ : "R3ÑR
px, y, zq ÞÑ x`2y`z@αPRα“Φppα, 0,0qq
Φpp1,0,´1qq “ p0,0,0q
Φ : "KrXs Ñ KrXs
PÞÑ P1
Ψ : "KrXs Ñ KrXs
PÞÑ XP
P“
n
ÿ
k“0
akXkP“Φ˜n
ÿ
k“0
ak
k`1Xk`1¸Φ Φp1q “ 0 Φ
@P, Q PKrXs,@λ, µ PK
ΨpλP `µQq “ XpλP `µQq “ λXP `µXQ “λΨpPq ` µΨpQq
Ψ
ΨpPq “ 0ñXP “0ñP“0
Ψ Ψ Q“1 Ψ @PPKrXs
XP “1ñX|1Q P Ψ
aqf:"R2ÑR2
px, yq ÞÑ px`y, x ´yqbqg:"RrXs Ñ RrXs
PÞÑ P`P1

F G E
@~u PE, D!p~u1, ~u2q P FˆG ~u “~u1`~u2
F G
p:"EÑE
~u ÞÑ ~u1
~u2
~u1“pp~uq
~u
F
G
F G
E
E“F‘G p F G
"@~u PF pp~uq “ ~u
@~u PG pp~uq “ 0
p
~
i,~ j, ~ kqR3
p:"R3ÑR3
px, y, zq ÞÑ px, y, 0q
Vectp
~
i,~ jqR~ k
pPLpEq
pôp˝p“p
pImpKerp E “Kerp‘Imp
p˝p“p E “Ker p‘Im p
pðq‚ ~x PKer pXIm p
~x PIm pD~y PE ~x “pp~yq~x Pker p pp~xq “ 0
pp~xq “ p˝pp~yq “ pp~yq “ ~x
~x “~ 0 Ker p‘Im p
‚~x PE
~x “pp~xq
loomoon
PIm p
`~x ´pp~xq
loooomoooon
PKer p
E“Ker p‘Im pIm pKer p ~x ÞÑ pp~xqp
pñq p F G ~x PE
~x “~x1
loomoon
PF
`~x2
loomoon
PG
pp~xq “ ~x1p˝pp~xq “ pp~x1q “ ~x1p˝p“p

F G E
@~u PE, D!p~u1, ~u2q P FˆG ~u “~u1`~u2
F G
s:"EÑE
~u ÞÑ ~u1´~u2
~u1
~u2~u
´~u2sp~uq
F
G
F G
E
E“F‘G s F G
"@~u PF sp~uq “ ~u
@~u PG sp~uq “ ´~u
p
~
i,~ j, ~ kqR3
s:"R3ÑR3
px, y, zq ÞÑ px, y, ´zq
Vectp
~
i,~ jqR~ k
sPLpEq
sôs˝s“IdE
"CÑC
zÞÑ zR C
Φ"KrXs Ñ KrXs
PpXq ÞÑ Pp´Xq
R@zPCpzq “ z z “z
zPRz“ ´z z PiRsRiR
Φ
EK