< K2 ç ¡ EUCLID M È Ñ

(1)

< K2 ç ¡ EUCLID M È Ñ

^

ïrB

z

<

l

<Æ>h:r, Ñþ6 xC $, §<Æ½¨, 2003

(2)

+ ä

q

Euclid /BNçß Eⁿ\"f ¿º &h s_ o\¦ Ô¦Ü¼Ð H affine ¨8`¦

½

+Ë1lx¨8s ôÇ. 7£¤,½+Ë1lx¨8T : Eⁿ→ EⁿH e__ P, Q ∈ Eⁿ\ @/ # 6£§`¦ëß7á¤ôÇ:

(3.1) d(P, Q) = d(T (P ), T (Q))

¨î

's1lxÉr¿º &h s_ o Ô¦Ü¼Ð Ä»t÷&l M:ëH\, affine¨8\"f Ø's1lxÂÒìrÉr {9éß ]jü@ . 7£¤, 1 ¨8 LAëß Òqty ¦, e__ ¿º &h P, Q ∈ Eⁿ\ @/ #

x =−−→

P Q 7' x ∈ Vⁿ`¦×þ .

Õ

ªQ (3.1)Ér6£§`¦_pôÇ: |x| = |LA(x)|

(3)

+ ä

P 2.3

e

__ x ∈ Vⁿ\ @/ # |x| = |L^A(x)| 1 ¨8LAH f§¨8, 7£¤,

tAA = Is. (sQôÇ '§>= A\¦f§'§>=s ôÇ.)

û B' å

e

__ x, y ∈ Vⁿ\ @/ #

y = Ax, det A 6= 0 s

. ÕªQ

|x|²=< x, x >=^tx x

|y|²=< Ax, Ax >= ^t(Ax)(Ax) =^tx^tAAx (§F ¸ÀÓ) s

.

(4)

"f |y| = |x|sl 0AôÇ ¸| Ér

tx^tAAx =^tx x s

Ù¼Ð

tAA = I

`

¦%3H.

(5)

A =







a11 a12 a13

a21 a22 a23

a₃₁ a₃₂ a₃₃





{9 M:,^tAA = Isl 0AôÇ ¸| Ér

a_1ia_1j+ a_2ia_2j+ a_3ia_3j= δ_ij =







1, if i = j 0, if i 6= j

(6)

+ ä

P 2.4

A f§'§>=s 6£§s $íwnôÇ.

(1) ^tAA = I (2) ^tA = A⁻¹ (3) det A = ±1

û B' å

Skip

§F\ ¸ : ÄºgË: −→ Äº8£¤

(7)

+ ä

P 2.5

f

§¨8Ér f§ýa³ð>\¦ f§ýa³ð>Ð ¨8ôÇ.

û B' å

{e1, e2, · · · , en}\¦ f§l, L^A\¦ f§¨8s

< ei, ej >= δij

s

Ù¼Ð

< Ae_i, Ae_j> =^t(Ae_i)(Ae_j) =^te_i^tAAe_j=^te_iIe_j

=^te_ie_j =< e_i, e_j>= δ_ij .

Õ

ªQÙ¼Ð {Ae¹, Ae2, · · · , Aen}Ér f§ls.

(8)

+ ä

P 2.6

Euclid /BNçß E³_ f§ýa³ð> {O; e¹, e2, e3}\"f ½+Ë1lx¨8 T : P (x₁, x₂, x₃) 7−→ P⁰(x⁰₁, x⁰₂, x⁰₃) H6£§õ °ú .











x⁰₁= a11x1+ a12x2+ a13x3+ b1

x⁰₂= a21x1+ a22x2+ a23x3+ b2

x⁰₃= a31x1+ a32x2+ a33x3+ b3

s

¦, '§>= A = (a_ij)H f§'§>=s.

û B' å

affine ¨8õ&ño 2.3\ _K "îÑþ .

(9)

+ ä

q

E³\"f

det A = 1 ½+Ë1lx¨8TbLA\¦f]X½+Ë1lx¨8(îr1lx)s ¦, det A = −1 ½+Ë1lx¨8TbLA\¦çß]X½+Ë1lx¨8(ìøÍ)s ôÇ.

(10)

z : + 8 q ä # ¢ ® £# e . > ~ 0 ï Fq ^ @+ 8

E²\"f ¿º &h P = (x₁, x₂), Q = (y₁, y₂)\ @/ # 4OP Q_ &h SH

6£§õ °ú s ½¨½+É Ãº e. (P, Q e__ ìr\ 0Au½+É M:¸

$í wn)

r O → P → Q ìøÍr>~½Ó¾Ós S > 0

r O → P → Q r>~½Ó¾Ós S < 0

S = x1y2−1

2y1yy2 − 1

2x1x2−1

2(x1− y1)(y2− x2)

=1

2(x1y2− y1x2) = 1 2

x1 y1

x₂ y₂

(11)

E²_ ½+Ë1lx¨8 T :







x⁰₁= a₁₁x₁+ a₁₂x₂+ b₁ x⁰₂= a₂₁x₁+ a₂₂x₂+ b₂

\

@/ #

O⁰= T (O) = (b1, b2), P⁰= T (P ) = (x⁰₁, x⁰₂), Q⁰ = T (Q) = (y⁰₁, y⁰₂) s

4O⁰P⁰Q⁰_ &h S⁰H6£§õ °ú .

S⁰= 1 2

x⁰₁− b1 y₁⁰ − b1

x⁰₂− b2 y₂⁰ − b2

= 1 2

a₁₁ a₁₂ a11 a12

x₁ y₁ x2 y2

=(det A)S

det A = 1s S⁰= S (y+þA_ ~½Ó¾Ós Ô¦)

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+ ä

P 2.7∼2.9

§F p.60∼61 ÃÐ¸

(13)

¹M 2

A²\"f

A = (0, 0) A⁰ = (1, −2) B = (1, −2) → B⁰ = (0, −4)

C = (2, 1) C⁰= (3, −3) (§F ¸)

affine¨8Ér½+Ë1lx¨8?

(14)

b ô >T

A²_ affine¨8 T :







x⁰₁= ax1+ bx2+ m x⁰₂= cx1+ dx2+ n

7

£¤, a b c d

! x1

x₂

!

= x⁰₁− m x⁰₂− n

!

\

0A ýa³ð\¦@/{9





 m = 1 n = −2

, a b c d

! 1 2

−2 1

!

= 0 − 1 3 − 1

−4 + 2 −3 + 2

!

= −1 2

−2 −1

!

"f, A = a b c d

!

= 1 5

−1 2

−2 −1

! 1 −2 2 1

!

=

3 5

4 5

−⁴₅ ³₅

!

tAA = I (check!)sÙ¼Ð AH f§'§>=s.

Õ

ªQÙ¼Ð T H½+Ë1lx¨8s.

(15)

a :

@ : @' Ö << K 2-3

2 : /BN:x

3, 5, 6, 8 : ¸Z> õ]j

3 ¸| : Ô¦&h`¦¿º >h °úHª_ ½+Ë1lx¨8s#Q <Ê.

6

£§ âÄº ìøÍYV [O"î

< K2 ç ¡ EUCLID M  È Ñ