X Y Kf : X ! Y

x " X#!X ! #!Y

v " #!X $! f (x + v) # f (x) " #!Y

X = Y = K = R

x $! ax + b

f : X ! Y

x $! x + v

f : X ! Y Kx0 " X

Lx0 : v " #!X $! f (x0 + v) # f (x0) "#!Y

Lx x#!f

x " X v $! f (x +v) # f (x) #!X ! #!Y x " X

x0 L :#!X ! #!Y

L(v) = v $! f (x0 + v) # f (x0).

v " #!X

f (x + v) # f (x) = f!x0 +##!x0x + v

"# f

!x0 +##!x0x

",

f (x + v) # f (x) = f!x0 +##!x0x + v

"# f (x0)

#!f!x0 +##!x0x

"# f (x0)

"

= L(##!x0x + v) # L(##!x0x)

= L(##!x0x) + L(v) # L(##!x0x)

= L(v)v

f : X ! YK X = Y

f : X ! Y#!f :#!X ! #!Y #!f

f #1 : Y ! X

X K GA(X)

X K nX %=

An(K) K

x0 x1 xn Xf : An(K) ! X

(!1, . . . , !n) $! x0 +n#

i=1!i###!x0xi " X.

f x0

xn

&###!x0x1, . . . ,###!x0xn'

n f (0 + v) # f (0)f

f : Ad(K) ! An(K)

f (x) = Ax + b = A(x # 0) + (b # 0),

A n ( d b Kn

z = (0, . . . , 0) " Ad(K)

f (z+ v)# f (z) A : Kd ! Kn

f (z + v) # f (z) = Av f (z) = b # 0

f (z + v) = Av+ b,

f (x) = Ax + b Kd

X n Yd p0, p1, . . . , pn

X n + 1 q0, q1, . . . , qn Yf : X ! Y f (pi) = qi i = 0, . . . , n

X %= An(K)

An(K) 0, e1, . . . , en

ei Kn

Y f (x) =Ax + b A

q1 # q0 qn # q0 bq0 Ai, j A bi

b i = 1 . . . n

f (pi) = Aei + b =

$%%%%%%%%%%%%%%%%%%%%%%%&

A1,i + b1

A2,i + b2

Ad,i + bd

'((((((((((((((((((((((()

f (p0) = b Ai, j

S ) X = An(K)

df : X ! An#d(K)

S = {x " X : f (x) = 0}.

f : X ! An#d(K) {x " X :

f (x) = 0} X d

W S P S

S = P +W .

W * W #!XW * #!X

#!X = W +W *.

dim(W) = d dim(W *) = n # d x " X x # P

x # P = w +w*

w " W w* " W * L :#!X !

W * L(w+w*) = w* L ker L = W W *

W * %= Kn#d

An#d(K) Kn#d f : X ! An#d(K)

f (x) = 0 + L(x # P) " An#d(K)

0 Kn#d

f (x) = 0 ,- x # P " W

,- x " P +W

,- x " S,

S = {x " X : f (x) = 0}f : X ! An#d(K) S = {x "

X : f (x) = 0} x0 " S L :#!X = Kn ! Kn#d Lv = f (x0 +

v) # f (x0) W ) Kn

n # (n # d) = d x0 " S f (x0) = 0x0 + v S

f (x0 + v) = 0 ,- Lv = 0 ,- v " W ,

S = x0 +W W d

2 3A2(R) ax + by + c = 0 A3(R) ax + by + cz + d =

0 A3(R) A3(R) ! A2(R)

*++++,++++-

ax + by + cz + d = 0

a*x + b*y + c*z + d* = 0

f : X ! Y

X YY X

S ) X #!S p " XS = p + #!S #!f :

#!X ! #!Y f #!f (v) =f (x0 + v) # f (x0)

f (S) = { f (p + s) : s " #!S }

= { f (p + s) # f (p) + f (p) : s " #!S }

= {#!f (s) + f (p) : s " #!S }

= { f (p) +w : w " #!f (#!S ) ) #!Y }.

#!f #!f (#!S ) ) #!Y

f : X ! Y fA, B, C " X

" =##!AC :

##!AB A* = f (A) B* = f (B) C* = f (C)

###!A*C* :

###!A*B* = ",

f

f : An(K) ! Am(K)

f (x) = Fx + b F b

C = A+ "(B # A),

C* = F(A+ "(B # A)) + b

= FA+ b + "(FB # FA)

= (FA+ b) + "(FB + b # FA# b)

= A* + "(B* # A*),

C* # A* = "(B* # A*)

f : X ! YK .= 2

A, B, C " XA*, B*, C*

###!A*C* :

###!A*B* = ##!AC :

##!AB

f (A) = A* f (B) = B* f (C) = C* f

f O " X

#!f (v) = f (O + v) # f (O)

v " #!X #!f#!X ! #!Y

1 # " K v " #!X OO + v O + #v

(O + #v#O) : (O + v#O) = #,

( f (O + #v) # f (O)) : ( f (O + v) # f (O)) = #,

f (O + #v) = f (O) + #( f (O + v) # f (O),

#!f (#v) = f (O + #v) # f (O) = #( f (O + v) # f (O)) = ##!f (v).

v, w " #!X O + 2v O + v + w O + 2wX 2

((O + 2w) # (O + 2v)) : ((O + v+ w) # (O + 2v)) = 2

,- O + 2w = (O + 2v) + 2((O + v+ w) # (O + 2v)).

f (O + 2w) # f (O + 2v) = 2 ( f (O + v+ w) # f (O + 2v)) ,

,- f (O + 2w) # f (O) # ( f (O + 2v) # f (O))

= 2 ( f (O + v+ w) # f (O) # ( f (O + 2v) # f (O))) ,

,- #!f (2w) # #!f (2v) = 2#!f (v+ w) # 2#!f (2v)

,- 2#!f (v+ w) = #!f (2v) + #!f (2w) = 2#!f (v) + 2#!f (w)

,- #!f (v+ w) = #!f (v) + #!f (w),

#!f f

d / 2f : Ad(R) ! Ad(R)

f

Ad(K) K .= R

K = R

$ : K ! K

z $! z C

K = C f : A2(K) ! A2(K)

f (x, y) = (x, y).

A = (a1, a2) B = (b1, b2) C = (c1, c2)

det

$%%%%%%%&b1 # a1 c2 # a1

b2 # a2 c2 # a2

'((((((() = 0,

fL : C2 ! C2 f (x0 + v) # f (x0)

L(v1, v2) = (v1, v2)

C

L(i, 0) = (#i, 0) .= (i, 0) = iL(1, 0) = L(i, 0).

S, T ) XX #!S ) #!T #!T ) #!S S 0 T

S T S 1 T .= 2

S ) X T ) XS 1 T .= 2 S ) T T ) S

P " S 1 T S T dim(S) 3dim(T) V ) W V W S T

x " S x # P " V x # P " W x " TS ) T

S ) X T ) Xdim(S) = dim(T) S 1 T .= 2 S = T

V = WS = T

#!S = #!T #!S ) #!TS T

A3(R)

A4(R)

S ) X x " XX T ) X dim(S)

x S

T * T xS

x " T 1 T *

X = A2(R)

S, T ) X

S, T ) X XS 1 T X

dim(S) + dim(T) 3 dim(X) + dim (S 1 T)

dim(#!S + #!T ) = dim(

#!X )

x0 " S 1 T

S = x0 +#!S

T = x0 +#!T

S 1 T = x0 +#!S 1 #!T ,

S 1 T ####!S 1 T = #!S 1 #!T

dim(#!S ) + dim(

#!T ) = dim(#!S + #!T ) + dim(

#!S 1 #!T ),

dim(S) + dim(T) = dim(#!S + #!T ) + dim(S 1 T)

3 dim(X) + dim(S 1 T),

dim(#!S + #!T ) 3 dim(

#!X ) = dim(X)

S, T ) X X #!S + #!T =#!X S 1 T

xS xT S T x " XS 1 T v " #!S w " #!T

x = xS + v = xT +w,

v " #!S w " #!T

xT # xS = v #w.

#!S + #!T = #!X xT # xS "#!X s " #!S t " #!T

xT # xS = s + t.

v = s " #!S w = #t " #!T v w

S ) X S .= XW ) #!X #!S +W =

#!XX S W pS,W : X ! S

x " XT = Tx,W x W S 1 Tx,W

#!S + ###!Tx,W =#!S +W =

#!Xdim(S 1 Tx,W ) = 0

S W pS,W

4x " X, pS,W (x) " S 1 Tx,W .

rS,W : X ! XS W ) #!X #!S +W = #!X

rS,W (x) = pS,W (x) +########!xpS,W (x)

r2 = r 5 r = 1X SS

f =pS,W S X W#!X #!S x " S

f (x) = x x " X L :#!X ! #!S

L(v) = f (x + v) # f (x) v {pS,W (x + v)} =S 1 Tx+v,W {pS,W (x)} = S 1 Tx,W Tx,W Tx+v,W

W x x + vx " S f (x) = x #!X = #!S +Wv " #!X v = s + w s " #!S w " W w " W

y " X W yy +w

y +W = y +w +W ,

Tx+v,W = Tx+s+w,W = Tx+s,W ,

f (x + v) = f (x + s) x " S s " #!S x + s " Sf (x + s) = x + s

f (x + v) # f (x) = f (x + s) # f (x) = x + s # x = s,

L(v) = s L :#!X = #!S +W ! #!S

r = rS,W

X ! X

r(x) = p(x) +####!xp(x) = p(x) + (p(x) # x)

p x " XL(v) = r(x + v) # r(x)

L(v) = p(x + v) + (p(x + v) # (x + v)) # (p(x) + (p(x) # x))

= p(x + v) # p(x) + (p(x + v) # p(x) # (x + v) + x)

= 2(p(x + v) # p(x)) # v,

v p(x + v) # p(x)v

K = R K = C

V %= Kn

X #!X %= Kn

x0 " X X#!X ! X

v $! x0 + v.

x0

Kn

0

A3(K) x, y, z (#, %,#1)(#, %,#1) z = 0 (x, y, z)

*+++++++,+++++++-

$%%%%%%%%%%%%%%&

xyz

'(((((((((((((()+ t

$%%%%%%%%%%%%%%&

#

%

#1

'(((((((((((((()

: t " K

.+++++++/+++++++0

z = 0

p :

$%%%%%%%%%%%%%%&

xyz

'(((((((((((((()$!

$%%%%%%%&x + #zy + %z

'((((((() =

$%%%%%%%&1 0 #

0 1 %

'((((((()

$%%%%%%%%%%%%%%&

xyz

'(((((((((((((().

A4(K) x, y, z, ww = 0 (a, b, c,#1)

A3(K) x, y, z w = 0 z = 0(#, %,#1)

*++++++++++,++++++++++-

$%%%%%%%%%%%%%%%%%%%%%%&

xyzw

'(((((((((((((((((((((()

+ t

$%%%%%%%%%%%%%%%%%%%%%%&

abc#1

'(((((((((((((((((((((()

: t " K

.++++++++++/++++++++++0

1{w = 0}

$%%%%%%%%%%%%%%%%%%%%%%&

xyzw

'(((((((((((((((((((((()

$!

$%%%%%%%%%%%%%%&

x + awy + bwz + cw

'(((((((((((((()=

$%%%%%%%%%%%%%%&

1 0 0 a0 1 0 b0 0 1 c

'(((((((((((((()

$%%%%%%%&1 0 #

0 1 %

'((((((()

$%%%%%%%%%%%%%%&

1 0 0 a0 1 0 b0 0 1 c

'(((((((((((((()=

$%%%%%%%&1 0 # a + #c0 1 % b + %c

'((((((()

A4(K)

z = w = 0 (a, b, c,#1)(#, %,#1, 0) (x, y, z, w) X, Y , Z , W

*+++++++++++++++++++++,+++++++++++++++++++++-

X = x + sa + t#

Y = y + sb + t%

Z = z + sc # t

W = w # s

W = 0

Z = 0

*++++++++++++,++++++++++++-

X = x + sa + t#

Y = y + sb + t%

s = w

t = z + sc

,-

*++++++++++++,++++++++++++-

X = x +wa + (z +wc)#

Y = y +wb + (z +wc)%

s = w

t = z +wc

X = x + #z + (a + #c)w

Y = y + %z + (b + %c)w.

A B

C D

E F

G H

I J

K L

M N

O P

AEMI BFNJ CGOK DHPL ACKI BDLJ EGOM FHPN ACGEBDHF IKOM JLPN ABJI CDLK EFNM GHPO ABFE CDHG IJNM KLPOABDC EFHG IJLK MNPO

p1 p2 Xp = !1 p1 + !2 p2

!1 + !2 = 1 K = R

!1 = !2 = 1/2p1 p2

##!p1p = ##!pp2

p1 p2

!1 / 0 !2 / 0 !1 + !2 = 1p = !1p1 + !2 p2.

[0, 1] ) R

S ) X = An(R)

W #!S #!X #!S +W =#!X p

S W r SW x " X p(x)

x r(x)

r SS x " S r(x) = x

A3(R) A2(R)$%%%%%%%%%%%%%%&

000

'(((((((((((((()$!

$%%%%%%%&01

'((((((()

$%%%%%%%%%%%%%%&

100

'(((((((((((((()$!

$%%%%%%%&10

'((((((()

$%%%%%%%%%%%%%%&

010

'(((((((((((((()$!

$%%%%%%%&11

'((((((()

$%%%%%%%%%%%%%%&

111

'(((((((((((((()$!

$%%%%%%%&00

'((((((()

A2(R) ! A2(R)

x = y

A3(R)$%%%%%%%%%%%%%%&

100

'(((((((((((((()

$%%%%%%%%%%%%%%&

1/21/20

'(((((((((((((()

$%%%%%%%%%%%%%%&

1/31/31/3

'(((((((((((((()

A4(R)

A4 3

A4(C)

r$%%%%%%%%%%%%%%&

xyz

'(((((((((((((()= t

$%%%%%%%%%%%%%%&

111

'(((((((((((((()

$%%%%%%%%%%%%%%&

xyz

'(((((((((((((()= t

$%%%%%%%%%%%%%%&

100

'(((((((((((((()

$%%%%%%%%%%%%%%&

xyz

'(((((((((((((()=

$%%%%%%%%%%%%%%&

010

'(((((((((((((()+ t

$%%%%%%%%%%%%%%&

001

'(((((((((((((()

A3(R)$%%%%%%%%%%%%%%&

xyz

'(((((((((((((()=

$%%%%%%%%%%%%%%&

110

'(((((((((((((()+ t

$%%%%%%%%%%%%%%&

001

'(((((((((((((()

A4(R)

x1, x2, x3, x4

*++++,++++-

x1 # x3 = 1

x2 # x4 = 1

$%%%%%%%%%%%%%%%%%%%%%%&

x1

x2

x3

x4

'(((((((((((((((((((((()

=

$%%%%%%%%%%%%%%%%%%%%%%&

1100

'(((((((((((((((((((((()

+ u

$%%%%%%%%%%%%%%%%%%%%%%&

1111

'(((((((((((((((((((((()

+ v

$%%%%%%%%%%%%%%%%%%%%%%&

1001

'(((((((((((((((((((((()

.

k A3(R)$%%%%%%%%%%%%%%&

21k

'(((((((((((((()

$%%%%%%%%%%%%%%&

1k0

'(((((((((((((()

$%%%%%%%%%%%%%%&

110

'(((((((((((((()

X f : X ! XABC

A*B*C* ##!AC :##!AB =

###!A*C* :

###!A*B*

ABC

GA(n, R) X = An(R) A " An(R)

A GA(n, R) A XGA(n, R) X ( X

f (A, B) = ( f (A), f (B)) f " GA(n, R) (A, B) " X2

(A, B) " X2 (A, B)GA(n, R) X ( X ( X

f (A, B, C) = ( f A, f B, f C) f " GA(n, R) A, B, C " X(A, B, C) " X3

(A, B, C)

A3(R) ! A2(R)

A3(R) r1

$%%%%%%%%%%%%%%&

xyz

'(((((((((((((()=

$%%%%%%%%%%%%%%&

100

'(((((((((((((()+ t

$%%%%%%%%%%%%%%&

010

'(((((((((((((()

r2

$%%%%%%%%%%%%%%&

xyz

'(((((((((((((()=

$%%%%%%%%%%%%%%&

010

'(((((((((((((()+ t

$%%%%%%%%%%%%%%&

001

'(((((((((((((()

r3

$%%%%%%%%%%%%%%&

xyz

'(((((((((((((()=

$%%%%%%%%%%%%%%&

001

'(((((((((((((()+ t

$%%%%%%%%%%%%%%&

100

'(((((((((((((()

A : A3(R) ! A3(R) A(r1) = r2 A(r2) =

r3 A(r3) = r1

f : X ! Y T ) YY f #1(T) X

A, B KP A B

##!AP :##!AB + ##!BP :

##!BA = 1

(##!AP :

##!AB) · (##!AB :##!AP) = 1

##!PA :##!PB = #

##!AP :##!AB

##!BP :##!BA

A, B, C Q(!0, !1, !2) A, B, C

Q = !0A+ !1B + !2C,

!0 + !1 + !2 = 1 Q .= A !1 + !2 .= 0Q* = 1

!1 + !2(!1B + !2C) BC QA

!1 + !2 = 0

A B CK PAB PBC PCA AB BC

CA A B CAPBC BPCA CPAB

####!APAB####!PABB

####!BPBC####!PBCC

####!CPCA####!PCAA

= 1

A B CK PAB PBC PCA

AB BC CA A B CPAB PBC PCA

####!APAB####!PABB

####!BPBC####!PBCC

####!CPCA####!PCAA

= #1.

$ : R ! R

x, y " R $(xy) = $(x)$(y) $(x + y) = $(x) + $(y)

k " Z $(k) = k

q " Q $(q) = q

x < y $(x) < $(y) t = y # x0 < $(

6t)2 = $(t)

$ R ! R

x " R $(x) = x $ 1R

X %= A2(K) K f : X !X

A B C X rA + x##!AB ##!AC s A + y##!AC##!AB r s A+ x##!AB + y##!AC " X

D = A + ##!AB + ##!AC (x, y) " K2 t A + x##!ADD A+ y##!AB t

AB A+ xy##!AB

f X Xf

f : X ! X

f (A) = A* f (B) = B* f (C) = C* A*B*C*

X A*B*C*

f g : X ! Xf (A) = A* = A f (B) = B* = B f (C) = C* = C

#!f : K2 ! K2 #!f (v) = f (A + v) # f (A)#!f (v+ w) = #!f (v) + #!f (w)

$ & f (A + x##!AB) = A + $(x)##!AB f (A +y##!AC) = A+ &(y)##!AC #!f (x, y) = ($(x),&(y))

OD $(x) = &(x) x " K$ : K ! K

(x, y) " K2 $(xy) = $(x)$(y) $(x + y) = $(x) + $(y)

K = R $(x) $(x) = x

f : Ad(R) ! Ad(R)