< K2 ç ¡ EUCLID M È Ñ

(1)

< K2 ç ¡ EUCLID M È Ñ

^

ïrB

z

<

l

<Æ>h:r, Ñþ6 xC $, §<Æ½¨, 2003

(2)

E²_ ½+Ë1lx¨8 M :







x⁰₁= a1x1+ a2x2+ c1

x⁰₂= b1x1+ b2x2+ c2

, (A = a₁ a₂ b1 b2

!

H f§'§>=)

\

@/ #, L : y₁

y2

!

= A x₁ x2

!

, T : x⁰₁ x⁰₂

!

= y₁ y2

! + c₁

c2

!

, M = T L (L : f§¨8, T : ¨î's1lx)s.

¢

¸ôÇ, L_ Ô¦&h`¦ (x1, x2) 6= (0, 0)s x₁

x2

!

= a₁ a₂ b1 b2

! x₁ x2

!

, a₁− 1 a₂ b1 b2− 1

! x₁ x2

!

= 0 0

! .

"f det(A − I) = 0.

(3)

I. det A = 1 ( ¸+ « כ Ö ò 6 B_ @Z 9 ) G B C I,

(1)L_ Ô¦&hs >rF HâÄº

det(A − I) = (a₁− 1)(b₂− 1) − a₂b₁= a₁b₂− a₂b₁− a₁− b₂+ 1

= det A − a1− b2+ 1 = 2 − a1− b2= 0

∴ a¹+ b2= 2 ———— `

tAA = IsÙ¼Ð, a²₁+ b²₁= a²₂+ b²₂= 1 ———— a

∴ a¹≤ 1, b2≤ 1 ———— b

"f,







a1= b2= 1 (`, b\ _K) b1= a2= 0 (a\ _K)

∴ A = I.

Õ

ªQÙ¼Ð, LÉr½Ó1px¨8s¦,M = T L = T Ér¨î's1lxs.

(4)

I. det A = 1 ( ¸+ « כ Ö ò 6 B_ @Z 9 ) G B C I,

(2)L_ Ô¦&hs >rF t ·ú§HâÄº

tAA = IsÙ¼Ð,

A⁻¹= b2 −a2

−b1 a1

!

= a1 b1

a2 b2

!

= ^tA

∴ a¹= b2, a2= −b1 ———— c det(A − I) 6= −\"f a1+ b₂6= 2sÙ¼Ð ———— d

a1= b26= 1 (cü< d\ _K) ———— e

¢

¸ôÇ^tAA = IsÙ¼Ð a²1+ b²₁= 1 ———— a

∴ a1< 1, b₁≤ 1 (aü< e\ _K) a1= cos θ, b1= sin θ (0 < θ < 2π)s

A = cos θ − sin θ sin θ cos θ

! s

.

Õ

ªQÙ¼ÐM = T LÉry θëßpu_ rõ ¨î's1lx_ ½+Ë$ís.

(5)

II. det A = −1 (µ ÿ + « כ Ö ò 6 B_ @Z 9 ) G B C I,

(1)L_ Ô¦&hs >rF HâÄº

det(A − I) = det A − a1− b2+ 1 = −a1− b2= 0

∴ b2= −a₁———— `

tAA = IsÙ¼Ð

a²₁+ b²₁= a²₂+ b²₂= 1, a1a2+ b1b2= 0 ———— a 7

£¤, b²₁= a²₂, a₁a₂− b₁a₁= 0 (`õ a\ _K)

∴ a²= ±b1

(6)

II. det A = −1 (µ ÿ + « כ Ö ò 6 B_ @Z 9 ) G B C I,

case1) a₂= −b₁s, A = a1 a2

−a2 −a1

! s

¦, b1b2= (−a2)(−a1) = a1a2.

a\ _K, a¹a2= 0sÙ¼Ð a¹= 0¢¸H a2= 0s.

a₁= 0s, det A = a²₂≥ 0s ÷&#Q ¸íHsÙ¼Ð a16= 0, a2= 0 7

£¤, A = a₁ 0 0 −a1

! s

¦, sכ ÉrA case2)_ :£¤Z>ôÇ âÄºs.

(7)

II. det A = −1 (µ ÿ + « כ Ö ò 6 B_ @Z 9 ) G B C I,

case2) a₂= b₁s, A = a1 a2

a2 −a1

! s

Ù¼Ð,

a²₁+ a²₂= a²₁+ b²₁= 1 (a\ _K)s.

s

M:,

a₁= cos θ, a₂= sin θ

Z~Ü¼

A = cos θ sin θ sin θ − cos θ

! s

.

(8)

II. det A = −1 (µ ÿ + « כ Ö ò 6 B_ @Z 9 ) G B C I,

w L_ Ô¦&hs, w = Aws¦, e__ a ∈ R\ @/ # aw = a(Aw) = A(aw)sÙ¼Ð aw¸ Ô¦&hs.

∴ ÕªaË>_ "é¶&h`¦tH f x2= x1tan φ

H¨8L\ @/ # Ô¦s.

#

l"f w = (cos φ, sin φ) ¿º, Aw = wsÙ¼Ð,

(9)

II. det A = −1 (µ ÿ + « כ Ö ò 6 B_ @Z 9 ) G B C I,

cos θ sin θ sin θ − cos θ

! cos φ sin φ

!

= cos θ cos φ + sin θ sin φ sin θ cos φ − cos θ sin φ

!

= cos φ sin φ

!

∴ cos(θ − φ) = cos φ, sin(θ − φ) = sin φ 7

£¤ θ − φ = φsÙ¼Ð,φ =θ 2

"f, ¨8LÉr f x₂= x1tan^θ₂\'aôÇ ìøÍs.

Õ

ªQÙ¼ÐM = T LÉrìøÍü< ¨î's1lx_ ½+Ë$ís.

(10)

II. det A = −1 (µ ÿ + « כ Ö ò 6 B_ @Z 9 ) G B C I,

(2)L_ Ô¦&hs >rF t ·ú§HâÄº

det(A − I) = −a₁− b₂6= 0 (p.67\ ¸)\"f a16= −b₂s¦,

tAA = IÐÂÒ' A⁻¹ = ^tAsÙ¼Ð A⁻¹= b2 −a2

−b1 a1

!

= a1 b1

a2 b2

!

= ^tA s

. 7£¤, a1= −b2 ÷&#Q ¸íHs.

Õ

ªQÙ¼Ð,çß]X½+Ë1lx¨8s"f Ls Ô¦&h`¦°út ·ú§HâÄºH

>rF t ·ú§H.

(11)

s

©`¦ 7áx½+Ë 6£§`¦%3H.

+ ä

P 2.10

E²\"f

(1) f]X½+Ë1lx¨8Érrõ ¨î's1lx_ ½+Ë$í







x⁰₁= x₁cos θ − x₂sin θ + c₁ x⁰₂= x₁sin θ + x₂cos θ + c₁

(0 ≤ θ ≤ 2π) (2) çß]X½+Ë1lx¨8ÉrìøÍü< ¨î's1lx_ ½+Ë$í







x⁰₁= x1cos θ + x2sin θ + c1

x⁰₂= x1sin θ − x2cos θ + c1

(0 ≤ θ ≤ 2π)

(12)

+ ä

P 2.11

§F ÃÐ¸(p.67)

(13)

+ ä

P 2.12

½

+Ë1lx¨8M : x⁰ = Ax + b (b 6= 0)_ Ô¦&h`¦ w .

(1) Ms f]X½+Ë1lx¨8s

w = (I − A)⁻¹b, x⁰ = A(x − w) + w (2) Ms çß]X½+Ë1lx¨8s

w = ₂^b, x⁰= A(x − w) + w

(14)

û B' å

(1) w Ô¦&hsÙ¼Ð w = Aw + bs¦, b = (I − A)ws.

A = a −b b a

!

Z~. ëß det(I − A) = 0s

(1 − a)²+ b²= a²+ b²− 2a + 1 = 2 − 2a = 0sÙ¼Ð a = 1, b = 0s.

7

£¤, A = Is¦ x⁰= x + b (b 6= 0)sÙ¼Ð ¨8M Ér¨î's1lxÜ¼Ð Ô

¦&hs >rF t ·ú§H. sכ Ér&ñ\ ¸íHs.

"f det(I − A) 6= 0 sÙ¼Ð

w = (I − A)⁻¹b.

¢

¸ôÇ x⁰= Ax + b = Ax + (I − A)wsÙ¼Ð x⁰ = A(x − w) + w s

.

(15)

(2) w Ô¦&hsÙ¼Ð w = Aw + bs.

Ms çß]X½+Ë1lx¨8sÙ¼Ð A = cos θ sin θ

sin θ − cos θ

!

A²= cos²θ + sin²θ 0 0 sin²θ + cos²θ

!

= 1 0 0 1

!

= I.

w = Aw + b = A(Aw + b) + b = w + Ab + b

∴ Ab + b = 0 A(b

2) + b = 1

2(Ab + b) + b 2 = b

2 sÙ¼Ð MÉr Ô¦&hw = b

2\¦°úH.

(16)

ô

Ç¼#, x⁰− w = A(x − w)sÙ¼Ð

x⁰ = A(x − w) + w.

(17)

+ ä

q

T²= I\¦ëß7á¤ H¨8T \¦@/½+Ës ôÇ. 7£¤, T = T⁻¹.

(18)

+ ä

P 2.13

@

/½+Ë ½+Ë1lx¨8Ér

ìøÍ, &hìøÍ, ½Ó1px¨8

÷

rs.

û B' å

(1) f]X½+Ë1lx¨8M : x⁰ = Ax + b (det A = 1) @/½+Ës x = Ax⁰+ b = A(Ax + b) + b = A²x + Ab + b ∀x

∴ A²= I, Ab + b = 0 (7£¤, (A + I)b = 0) i) b = 0 âÄº, A²= I = ^tAA ( ∵ AH f§'§>=)sÙ¼Ð A = ^tA

cos θ − sin θ sin θ cos θ

!

= cos θ sin θ

− sin θ cos θ

!

sin θ = 0 7£¤, θ = 0¢¸H θ = π.

"f A = I ¢¸H A = −I.

(19)

ii) b 6= 0 âÄº, (A + I)b = 0sÙ¼Ð (det(A + I) = 0`¦ f]X >íß A = −I\¦%36£§)

A = −I 7£¤, x⁰= −x + b.

Õ

ªQÙ¼Ð b 6= 0s, M ÉrÔ¦&h b

2\'aôÇ &hìøÍs.

(20)

(2)çß]X½+Ë1lx¨8M : x⁰ = Ax + b (det A = −1) @/½+Ës x = Ax⁰+ b = A(Ax + b) = A²x + Ab + b ∀x

∴ A²= I, Ab + b = (A + I)b = 0

¢

2\¦°ú¦ MÉr@/½+ËsÙ¼Ð x = Ax⁰+ b

= A(Ax − Aw + w) + b

= A²x − A²w + w + b

= x − w + w + b 7

£¤, b = 0s.

Õ

ªQÙ¼Ð MÉr&ño 2.10\ _ #f x₂= x₁tan^θ₂\'aôÇ

ìøÍs.

(21)

E²\"f ¿º f m, n\'aôÇ ìøÍ\¦yy R^m, Rns ½+Ë$í¨8 R_nR_mÉr6£§õ °ú .

(1) m//ns RnRmÉr¨î's1lxs.

(2) m \// ns Õª §&h\'aôÇ rs.

û B' å

Rm: x⁰ = Ax + b (det A = −1, A²= I) Rn: x⁰⁰= Bx⁰+ c (det B = −1, B²= I)

RnRm: x⁰⁰= B(Ax + b) + c = BAx + Bb + c det(BA) = det B · det A = 1

(22)

(1) BA = Is A = BA · A = B · I = BsÙ¼Ð m//ns¦ s M:, x⁰⁰= x + Bb + csÙ¼Ð

RnRmÉr¨î's1lxs.

(2) BA 6= Is A 6= B⁻¹= BsÙ¼Ð m \// ns¦ s M:, Ô

¦&h w\¦°úH.

w = BAw + Bb + csÙ¼Ð (I − BA)w = Bb + c s. 7£¤, w = (I − BA)⁻¹(Bb + c) (&ño 2.12(1)) Õ

ªQÙ¼Ð RnR_mÉr Ô¦&h w\'aôÇ rs.

(23)

¹M 2(1)q ; d< K

Rm: x⁰ y⁰

!

= 0 1 1 0

! x y

! + −1

1

!

Rn: x⁰⁰ y⁰⁰

!

= 0 1 1 0

! x⁰ y⁰

! + −2

2

!

_

Ô¦&h[þtõ RnR_m`¦¸KÐ.

(24)

a :

@ : @' Ö << K 2-4

1, 2 : /BN:x

3, 5, 6 : ¸Z> õ]j

< K2 ç ¡ EUCLID M  È Ñ