Ds Va Hhgt - Thuvientoanhoc

8/6/2019 Ds Va Hhgt - Thuvientoanhoc

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A I S O V A H N H H O C G I A I T C H 1 - 2

G i a o t r n h a i h o c a i c n g N g a n h T o a n - T i n h o c

T a L e L i

- a i H o c a l a t -

- 2 0 0 5 -


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a i s o v a H n h h o c g i a i t c h 1 - 2

T a L e L i

M u c l u c

P h a n I :

C h n g 0 . K i e n t h c c h u a n b

1 . C a c c a u t r u c a i s o c b a n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 . T r n g s o p h c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 . a t h c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

C h n g I . K h o n g g i a n v e c t o r h n h h o c

1 . V e c t o r h n h h o c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5

2 . C s D e s c a r t e s - T o a o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7

3 . C o n g t h c a i s o c u a c a c p h e p t o a n t r e n v e c t o r . . . . . . . . . . . . . . . . . . . . . . . . . 1 9

4 . n g t h a n g v a m a t p h a n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2

C h n g I I . M a t r a n - P h n g p h a p k h G a u s s

1 . M a t r a n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7

2 . C a c p h e p t o a n t r e n m a t r a n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 8

3 . P h n g p h a p k h G a u s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5

C h n g I I I . K h o n g g i a n v e c t o r

1 . K h o n g g i a n v e c t o r - K h o n g g i a n v e c t o r c o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1

2 . C s - S o c h i e u - T o a o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4

3 . T o n g - T c h - T h n g k h o n g g i a n v e c t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 9

C h n g I V . A n h x a t u y e n t n h

1 . A n h x a t u y e n t n h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3

2 . A n h x a t u y e n t n h v a m a t r a n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 8

3 . K h o n g g i a n o i n g a u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2

C h n g V . n h t h c

1 . n h t h c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5

2 . T n h c h a t c u a n h t h c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 7

3 . T n h n h t h c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 9

4 . M o t s o n g d u n g c u a n h t h c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3


3/156

P h a n I I :

C h n g V I . C h e o h o a

1 . C h u y e n c s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1

2 . V e c t o r r i e n g - G a t r r i e n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4

3 . D a n g n g c h e o - C h e o h o a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5

C h n g V I I . K h o n g g i a n v e c t o r E u c l i d

1 . K h o n g g i a n v e c t o r E u c l i d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1

2 . M o t s o n g d u n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 8

3 . T o a n t t r c g i a o - M a t r a n t r c g i a o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 2

4 . T o a n t o i x n g - C h e o h o a t r c g i a o m a t r a n o i x n g . . . . . . . . . . . . . . . 1 0 9

C h n g V I I I . D a n g s o n g t u y e n t n h - D a n g t o a n p h n g

1 . D a n g s o n g t u y e n t n h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 3

2 . D a n g t o a n p h n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 4

3 . D a n g c h n h t a c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 5

C h n g I X . A p d u n g v a o h n h h o c

1 . C a u t r u c a f f i n c h n h t a c c u a m o t k h o n g g i a n v e c t o r . . . . . . . . . . . . . . . . . . . . 1 2 5

2 . M o t s o a n h x a a f f i n t h o n g d u n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 8

3 . n g , m a t b a c 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 3

B a i t a p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 9


4/156

0 . K i e n t h c c h u a n b

C h n g n a y n e u n h n g h a v e c a c c a u t r u c a i s o c b a n l a n h o m , v a n h v a t r n g .

P h a n t i e p t h e o l a m o t s o k i e n t h c t o i t h i e u v e s o p h c v a a t h c .

1 . C a c c a u t r u c a i s o c b a n

1 . 1 n h n g h a . C h o A l a m o t t a p h p . M o t p h e p t o a n h a i n g o i t r e n A l a m o t

a n h x a :

: A A AK h i o a n h c u a c a p (x, y) A A b i a n h x a s e c k y h i e u l a x y

P h e p t o a n g o i l a c o t n h k e t h p n e u u 1 (x y) z = x (y z), x , y , z A P h e p t o a n g o i l a c o

t n h g i a o h o a n n e u u x y = y x, x, y A

P h a n t e A , g o i l a p h a n t n v , n e u u x e = e x = x, x AK h i

v i e t t h e o l o i c o n g + t h p h a n t n v g o i l a p h a n t k h o n g v a k y h i e u l a 0.

K h i

v i e t t h e o l o i n h a n t h p h a n t k y h i e u l a 1. G i a s p h e p t o a n c o p h a n t n v e . K h i o x A g o i l a k h a n g h c h n e u u t o n t a i

x A s a o c h o : x x = x x = e . K h i o x p h a n t n g h c h a o c u a x .K h i v i e t t h e o l o i c o n g , t h p h a n t n g h c h a o c u a x g o i l a

p h a n t o i v a k y h i e u

l a x . K h i v i e t t h e o l o i n h a n , t h p h a n t n g h c h a o c u a x k y h i e u l a x1 h a y 1x

.

N h a n x e t . P h a n t n v n e u c o l a d u y n h a t :

N e u e1, e2 l a h a i p h a n t n v , t h e1 = e1 e2 = e2 .N h a n x e t . N e u

c o t n h k e t h p , t h p h a n t n g h c h a o c u a

xn e u c o l a d u y n h a t :

N e u x, x

l a h a i p h a n t n g h c h a o c u a x

, t h x = xe = x(x x) = (xx)x =

e x = x .

B a i t a p : H a y x e t c a c p h e p t o a n c o n g v a n h a n t r e n A := N, Z, Q, R c o t n h c h a t g ? C o p h a n t n v ? C o p h a n t n g h c h a o ?

1 . 2 . N h o m . M o t n h o m l a m o t c a p (G, ), t r o n g o G l a m o t t a p h p k h o n g r o n g , c o n

l a m o t p h e p t o a n h a i n g o i t r e n G

, t h o a c a c i e u k i e n s a u :

( G 1 )

c o t n h k e t h p .

( G 2 ) c o p h a n t n v .

( G 3 ) M o i p h a n t c u a G e u c o p h a n t n g h c h a o .

N h o m G c g o i l a n h o m g i a o h o a n h a y n h o m A b e l n e u :

( G 4 ) c o t n h g i a o h o a n .

N g i t a t h n g n o i n h o m G

t h a y v (G, ) k h i a n g a m h i e u p h e p t o a n n a o . Q u i c n a y c u n g d u n g c h o k h a i n i e m v a n h , t r n g t i e p s a u .

1

T r o n g g i a o t r n h n a y :

n e u u

=

n e u v a c h n e u

.


5/156

2

V d u .

a ) T a p N v i p h e p c o n g k h o n g l a n h o m v k h o n g c h a p h a n t o i . T a p Z, Q, R l a n h o m g i a o h o a n v i p h e p c o n g , n h n g k h o n g l a n h o m v i p h e p n h a n v 0 k h o n g c o p h a n t n g h c h a o .

b ) T a p c a c s o n g a n h t m o t t a p X l e n c h n h X l a m o t n h o m v i p h e p h p a n h x a .

N o i c h u n g n h o m n a y k h o n g g i a o h o a n .

1 . 3 V a n h . M o t v a n h

l a m o t b o b a (R, +, ), t r o n g o R l a m o t t a p k h o n g r o n g , c o n + v a l a c a c p h e p t o a n t r e n R , t h o a c a c i e u k i e n s a u : ( R 1 ) (R, +) l a m o t n h o m g i a o h o a n . ( R 2 ) P h e p n h a n c o t n h k e t h p . ( R 3 ) P h e p n h a n c o t n h p h a n p h o i v e h a i p h a o i v i p h e p c o n g :

x(y + z) = xy + xz v a (y + z)x = yx + zx x , y , z R

N e u p h e p n h a n c o t n h g i a o h o a n t h R g o i l a v a n h g i a o h o a n

.

V d u .

a ) Z, Q, R v i p h e p c o n g v a n h a n l a c a c v a n h g i a o h o a n . b ) Zp c a c l p c a c s o n g u y e n o n g d t h e o m o t s o p l a v a n h g i a o h o a n v i p h e p c o n g v a n h a n c n h n g h a :

[m] + [n] = [m + n], [m][n] = [mn]

1 . 3 T r n g . M o t t r n g

l a m o t v a n h g i a o h o a n c o n v 1

= 0 v a m o i p h a n t

k h a c k h o n g c u a K e u c o p h a n t n g h c h a o . M o t c a c h a y u , m o t t r n g l a b o

b a (K, +, ), t r o n g o K l a t a p k h o n g r o n g , + v a l a c a c p h e p t o a n t r e n K t h o a 9 i e u k i e n s a u v i m o i x , y , z K:( F 1 ) (x + y) + z = x + (y + z)( F 2 ) 0 K, x + 0 = 0 + x = x( F 3 ) x K, x + (x) = x + x = 0( F 4 ) x + y = y + x( F 5 ) (xy)z = x(yz)( F 6 ) 1 K, 1 = 0, x1 = 1x = x( F 7 ) K h i x = 0, x1 K, xx1 = x1x = 1( F 8 ) xy = yx( F 9 ) x(y + z) = xy + xz

V d u .

a ) V a n h (Z, +, ) k h o n g l a t r n g . (Q, +, ), (R, +) l a c a c t r n g . b ) N e u p l a s o n g u y e n t o , t h Zp l a m o t t r n g . H n n a , Zp l a t a p h u h a n v a v i m o i [n] Zp , [n] + + [n]

pl a n

= [0].

a c s o c u a m o t t r n g K, k y h i e u c h a r (K), l a s o t n h i e n d n g b e n h a t s a o


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C h n g 0 . K i e n t h c c h u a n b 3

c h o 1 + + 1 n l a n

= 0. N e u k h o n g c o s o t n h i e n n h v a y , t h K g o i l a c o a c s o 0.

V d u . Q, R c o a c s o 0, c o n Zp c o a c s o p . T a c o 1 + 1 = 0 t r o n g Z2 !

2 . T r n g s o p h c

T r e n t r n g s o t h c , k h i x e t p h n g t r n h b a c h a i ax2 + bx + c = 0 t r n g h p b2 4ac < 0 p h n g t r n h v o n g h i e m v t a k h o n g t h e l a y c a n b a c h a i s o a m . e c a c p h n g t r n h n h v a y c o n g h i e m , t a c a n t h e m v a o t a p c a c s o t h c c a c c a n b a c h a i

c u a s o a m . P h a n n a y t a s e x a y d n g t a p c a c s o p h c C l a m r o n g t a p s o t h c R,t r e n o n h n g h a c a c p h e p t o a n c o n g v a n h a n e C l a m o t t r n g . H n n a , m o i p h n g t r n h b a c h a i , c h a n g h a n

x2 + 1 = 0 , e u c o n g h i e m t r o n g C.

2 . 1 n h n g h a . T a d u n g k y h i e u i

, g o i l a c s o a o , e c h n g h i e m p h n g t r n h

x2 + 1 = 0 , i . e . i2 = 1 . T a p s o p h c l a t a p d a n g : C = {z : z = a + ib, v i a, b R}z = a + ib g o i l a s o p h c , a = R e z g o i l a p h a n t h c , b = I m z g o i l a p h a n a o .

z1 = z2 n e u u R e z1 = R e z2, I m z1 = I m z2 .T a x e m R l a t a p c o n c u a C k h i o n g n h a t R = {z C : I m z = 0}T s o a o s i n h r a t v i e c n g i t a k h o n g h i e u c h u n g k h i m i p h a t h i e n r a s o p h c .

T h c r a s o p h c r a t t h c n h s o t h c v a y .

V d u .

a ) S o p h c z = 6 + i2 c o p h a n t h c R e z = 6, p h a n a o I m z = 2.b ) e g i a i p h n g t r n h

z2 + 2z + 4 = 0 , t a b i e n o i z2 + 2z + 4 = (z + 1)2 + 3 .V a y p h n g t r n h t n g n g (z + 1)2 = 3. S u y r a n g h i e m z = 1 i3.

S a u a y l a n h n g h a c a c p h e p t o a n v a t h c h i e n .

2 . 2 C a c p h e p t o a n . T r e n C c o h a i p h e p t o a n c n h n g h a n h s a u :

P h e p c o n g . (a + ib) + (c + id) = (a + c) + i(b + d)P h e p n h a n . (a + ib)(c + id) = (ac bd) + i(ad + bc)

N h a n x e t . P h e p n h a n c t n h n h n h a n c a c s o t h o n g t h n g v i c h u y l a i2 = 1.

M e n h e . V i c a c p h e p t o a n t r e n

Cl a t r n g s o .

M e n h e t r e n d e s u y t n h n g h a v i c h u y l a :

P h e p c o n g c o p h a n t k h o n g l a 0 = 0+ i0, p h a n t o i c u a z = a + ib l a z = a ib .P h e p n h a n c o p h a n t n v l a 1 = 1 + i0 , n g h c h a o c u a z = a + ib = 0 l a z1 =

1

z=

a

a2 + b2 i b

a2 + b2

S t o n t a i v a v i e c t m n g h c h a o c t h c h i e n b i p h e p c h i a

a + ib

c + id( c + id = 0)


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4

k h i g i a i p h n g t r n h a + ib = (c + id)(x + iy) . o n g n h a t p h a n t h c , p h a n a o t a c o

cx dy = adx + cy = b

V a y

a + ib

c + id=

ac + bd

c2 + d2+ i

bc adc2 + d2

P h e p l i e n h p . z = a ib g o i l a s o p h c l i e n h p c u a z = a + ib.

T n h c h a t . z = z , z1 + z2 = z1 + z2 , z1z2 = z1z2 .

N h a n x e t . N e u z = a + ib, t h zz = a2 + b2 . T o c o t h e c h i a 2 s o p h c b a n g c a c h n h a n s o l i e n h i e p c u a m a u , c h a n g h a n

2 5i3 + 4i

=(2 5i)(3 4i)(3 + 4i)(3

4i)

=6 23i + 20i2

32

42i2

=14 23i

25

2 . 3 B i e u d i e n s o p h c . S a u a y l a m o t s o b i e u d i e n k h a c n h a u c u a s o p h c

Bz

O E x

T

y

Ti

a

b

r

D a n g a i s o . z = a + ib, a, b R, i2 = 1.

D a n g h n h h o c . z = (a, b), a, b R.T r o n g m a t p h a n g a v a o h e t o a t r u c D e s c a r t e s v i 1 = (1, 0), i = (0, 1) l a 2 v e c t o r c s . K h i o m o i s o p h c z = a + ib c b i e u d i e n b i v e c t o r (a, b), c o n C c o n g n h a t v i R2 . T r o n g p h e p b i e u d i e n n a y p h e p c o n g s o p h c c b i e u t h b i p h e p c o n g v e c t o r h n h h o c .

D a n g l n g g i a c . z = r(cos + i sin )B i e u d i e n s o p h c z = (a, b) t r o n g t o a o c c (r, ) , t r o n g o r l a o d a i c u a z , l a g o c n h h n g t a o b i 1 = (1, 0) v a z t r o n g m a t p h a n g p h c . T a c o :

a = r cos b = r sin

v a

r = |z| = a2 + b2, g o i l a m o d u l c u a z = A r g z, g o i l a a r g u m e n t c u a z

V a y n e u z = 0, t h cos = a

a2 + b2, sin =

ba2 + b2

.

T a t h a y c o v o s o g i a t r s a i k h a c n h a u k2, k Z, t r o n g o c o m o t g i a t r (, ]


8/156


g o i l a g i a t r c h n h v a k y h i e u l a a r g z

. V a y c o t h e v i e t

A r g z = a r g z + k2, k Z.

V d u . z =

3 i c o m o d u l |z| =

(

3)2 + (1)2 = 2 , v a a r g u m e n t a r g z = 3

( s u y t tan = 13

v a R e z > 0 ) . V a y

3 i = 2(cos(

3) + i sin(

3)).

M o i c a c h b i e u d i e n s o p h c c o t h u a n t i e n r i e n g . S a u a y l a m o t s o n g d u n g .

2 . 4 M e n h e . |z1z2| = |z1||z2| v a A r g (z1z2) = A r g z1 + A r g z2S u y r a c o n g t h c d e M o i v r e

(r(cos + i sin ))n = rn(cos n + i sin n), n

N

C h n g m i n h : N e u z1 = r1(cos 1 + i sin 1), z2 = r2(cos 2 + i sin 2), t h

z1z2 = r1r2(cos 1 cos 2 sin 1 sin 2) + i(sin 1 cos 2 + cos 1 sin 2)= r1r2(cos(1 + 2) + i sin(1 + 2))

S u y r a |z1z2| = r1r2 = |z1||z2|, v a A r g (z1z2) = 1 + 2 + 2k = A r g z1 + A r g z2 .

N h a n x e t . V e m a t h n h h o c p h e p n h a n s o p h c r(cos + i sin ) v i s o p h c zl a p h e p c o d a n v e c t o r

zt s o

rv a q u a y g o c

. ( x e m h n h v e )

2 . 5 C a n b a c n c u a s o p h c . C h o z C v a n N. M o t c a n b a c n c u a z l a m o t s o p h c

wt h o a p h n g t r n h

wn = z . e g i a i p h n g t r n h t r e n , b i e u d i e n z = r(cos + i sin ) v a w = (cos + i sin ).T c o n g t h c d e M o i v r e n(cos(n) + i sin(n)) = r(cos + i sin ) .S u y r a

= n

r ( c a n b a c n t h e o n g h a t h c )

n = + 2k, k Z

V a y k h i z

= 0, p h n g t r n h c o u n g n n g h i e m p h a n b i e t :

wk =n

r(cos(

n+ k

2

n) + i sin(

n+ k

2

n)), k = 0, , n 1.

K h iz = 0, k y h i e u nz l a t a p n c a n b a c n c u a z . 0 = 0 .

V e m a t h n h h o c c h u n g l a c a c n h c u a m o t a g i a c e u n

c a n h , n o i t i e p n g t r o n

t a m 0 b a n k n h n

r .


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6

Bsz

OE

T

s

r(cos + i sin )z

N h a n r(cos + i sin ) v i z wn = 1, v i n = 8

sw

sw

sw

sw

s

s

s

s

V d u .

a ) C a n b a c n

c u a 1 l a n

s o p h c : 1, n, , n1n , v i n = cos 2n + i sin 2nb ) e t m c a c g a t r c u a

1 + i , t a b i e u d i e n 1 + i =

2(cos

4+ i sin

4).

S u y r a

1 + i = 2 (cos(

12

+ 2k3

) + i sin( 12

+ 2k3

)), k

Z.V a y c o 3 g i a t r p h a n b i e t l a :

k = 0, w0 = 2 (cos(12

) + i sin( 12

))

k = 1, w1 = 2 (cos(34

) + i sin(34

)) = 3w0

k = 2, w2 = 2 (cos(1712

) + i sin(1712

)) = 3w0

3 . a t h c

3 . 1 n h n g h a . C h o K l a m o t t r n g . M o t a t h c

t r e n K l a b i e u t h c d a n g

P(X) = a0 + a1X +

+ anX

n,

t r o n g o n N, v a ak K, k = 0, , n, g o i l a h e s o b a c k c u a P(X).

H a i a t h c g o i l a b a n g n h a u n e u u m o i h e s o c u n g b a c c u a c h u n g b a n g n h a u .

N e u an = 0, t h n g o i l a b a c c u a P(X) v a k y h i e u n = deg P(X) , an = l c P(X) .

N e u ak = 0 v i m o i k , t h P(X) g o i l a a t h c k h o n g v a q u i c deg(0) = .

T a t h n g v i e t d i d a n g t o n g : P(X) =n

k=0

akXk

h a y P =k

akXk

l a t o n g v o h a n

n h n g c h c o h u h a n ak = 0.

K y h i e u K[X] l a t a p m o i a t h c t r e n K.

3 . 2 C a c p h e p t o a n t r e n a t h c . T r e n K[X] c o h a i p h e p t o a n c o n g v a n h a n n h n g h a n h s a u :

P h e p c o n g :

k

akXk +

k

bkXk =

k

(ak + bk)Xk

P h e p n h a n : (i

aiXi)(j

bjXj) =

k

ckXk

v i ck = a0bk + + akb0 =

i+j=k

aibj .

M e n h e . K[X] l a v i h a i p h e p t o a n t r e n l a m o t v a n h g i a o h o a n .

B a i t a p : C h n g m i n h m e n h e t r e n .


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N h a n x e t . deg P(X)Q(X) = deg P(X) + deg Q(X), v i m o i P(X), Q(X) K[X].

3 . 3 P h e p c h i a E u c l i d . C h o h a i a t h c P0(X), P1(X) K[X], P1(X) = 0 .K h i o t o n t a i d u y n h a t c a c a t h c Q(X), R(X)

K[X] , s a o c h o

P0(X) = Q(X)P1(X) + R(X), deg R(X) < deg P1(X)

T a g o i Q(X) l a t h n g

, R(X) l a p h a n d

c u a p h e p c h i a P0(X) c h o P1(X) , v a c x a y d n g c u t h e t h e o t h u a t t o a n s a u :

T h u a t t o a n c h i a E u c l i d .

I n p u t :P0, P1 K[X], P1 = 0

O u t p u t :Q, R K[X], t h o a P0 = QP1 + R, deg R < deg P1 .

T r c h e t c h o R0 = P0, Q0 = 0.G i a s v o n g l a p t h k t a c o Qk, Rk

K[X] , t h o a P0 = QkP1 + Rk

N e u nk = deg Rk deg P1 < 0, t h a c h i a x o n g Q = Qk, R = RkN e u

nk = deg Rk deg P1 > 0, t h k h h e s o b a c c a o n h a t c u a Rk b a n g c a c h : Rk+1 = Rk l c (Rk)

l c (P1)XnkP1

Qk+1 = Qk +l c (Rk)

l c (P1)Xnk

T a c o P0 = Qk+1P1 + Rk+1D o deg Rk+1 < deg Rk , n e n e n v o n g l a p t h m deg P0 , t a c o deg Rm < deg P1 .K h i o Q = Qm, R = Rm .

V d u . T h u a t t o a n c h i a E u c l i d X4 2X3 6X2 + 12X+ 15 c h o X3 + X2 4X 4c o t h e t h c h i e n t h e o s o

R0 = X4 2X3 6X2 + 12X + 15 | X3 + X2 4X 4

R1 = 3X3 2X2 + 16X + 15 X 3R2 = X

2 + 4X + 3

V a y X4 2X3 6X2 + 12X + 15 = (X3 + X2 4X 4)(X 3) + X2 + 4X + 3

B a i t a p : T h c h i e n p h e p c h i a P(X) = a0 + a1X + + anXn c h o X c .

3 . 4 c c h u n g l n n h a t . a t h c P(X) K[X] g o i l a c h i a h e t c h o a t h c D(X)

K[X] n e u u t o n t a i a t h c A(X)

K[X] , s a o c h o P(X) = A(X)D(X).

K h i o D(X) g o i l a m o t c c u a P(X) v a k y h i e u D(X)|P(X) . c c h u n g l n n h a t c u a c a c a t h c P0(X), P1(X) K[X] , l a m o t a t h c D(X) K[X] , t h o a i e u k i e n :

D(X)|P0(X), D(X)|P1(X) v a n e u C(X)|P0(X), C(X)|P1(X) t h C(X)|D(X)

K h i o k y h i e u D(X) = G C D (P0(X), P1(X))

N h a n x e t . c c h u n g l n n h a t c x a c n h s a i k h a c m o t h a n g s o t l e .

N h a n x e t . N e u P0 = QP1 + R , t h G C D (P0, P1) = G C D (P1, R), v c c h u n g c u a


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8

P0, P1 l a c c h u n g c u a P1, R .

N g o a i r a G C D (R, 0) = R, n e n c c h u n g l n n h a t c x a y d n g t d a y p h a n d c u a t h u a t c h i a E u c l i d , n h s a u :

T h u a t t o a n t m G C D .

I n p u t : P0, P1 K[X], P0, P1 = 0O u t p u t : G C D (P0, P1) v a U, V K[X], t h o a U P0 + V P1 = G C D (P0, P1)

X a y d n g d a y a t h c k h a c k h o n g (P0, P1, P2, , Pm) , v i Pk l a p h a n d c u a p h e p c h i a Pk2 c h o Pk1 :

Pk2 = Qk1Pk1 + Pk (k = 2, , m 1)Pm2 = Qm1Pm1 + PmPm1 = QmPm

T h e o n h a n x e t t r e n t a c o G C D (P0, P1) = G C D (Pm, 0) = Pm

T h u a t t o a n c o n c h o c a c d a y a t h c (U0, , Um) v a (V0, , Vm) , t h o a Pk = UkP0 + VkP1, (k = 0, , m) ()

T r c h e t , k h i k = 0, 1, t a p h a i c o U0 = 1, V0 = 0 v a U1 = 0, V1 = 1. S a u o e q u i

Uk = Uk2 Qk1Uk1 v a Vk = Vk2 Qk1Vk1 (k = 2, m)T a k i e m t r a d a y t h o a () b a n g q u i n a p . G i a s () u n g e n k 1. K h i o

Pk = Pk2 Qk1Pk1= (Uk2P0 + Vk2P1) Qk1(Uk1P0 + Vk1P1)= (U

k2

Q

k1)P0 + (V

k2

Q

k1V

k1)P1

= UkP0 + VkP1

K h i U = Um, V = Vm t a c o U P0 + V P1 = Pm = G C D (P0, P1). V a y t a c o :

a n g t h c B e z o u t . C h o P0(X), P1(X) K[X] . K h i o t o n t a i U(X), V(X) K[X]s a o c h o

G C D (P0(X), P1(X)) = U(X)P0(X) + V(X)P2(X)

V d u . V i P0(X) = X

4 2X3 6X2 + 12X + 15 v a P1(X) = X3 + X2 4X 4 ,c a c b c c u a t h u a t t o a n t r e n c t h e h i e n q u a b a n g s a u

k Pk2 Pk1 Qk1 Pk2 X4 2X3 6X2 + 12X + 15 X3 + X2 4X 4 X 3 X2 + 4X + 33 X3 + X2 4X 4 X2 + 4X + 3 X 3 5X + 54 X2 + 4X + 3 5X + 5 1

5X + 3

50

5 5X + 5 0

U0 = 1, U1 = 0, U2 = 1, U3 = X + 3, U4 = 15X2 45 , U5 = X + 3

V0 = 0, V1 = 1, V2 = X+3, V3 = X26X+10, V4 = 15

X3+3

5X2+

3

5X3, V5 = X26X+10


12/156


13/156

1 0

( i i i ) N e u n = deg P(X) l a l e , t h P(X) c o n g h i e m t h c .

C h n g m i n h :

( i ) G i a s P(X) = a0 + a1X +

+ anX

n

R[X] . K h i o v i m o i c

C, t a c o

P(c) = a0 + a1c + + ancn = a0 + a1c + + ancn = P(c)( e y l a ak R n e n ak = ak , v a l i e n h p c u a t o n g ( t c h ) l a t o n g ( t c h ) l i e n h p ) V a y P(c) = 0 k h i v a c h k h i P(c) = 0. S u y r a ( i ) . ( i i ) C h o c = a + ib C. K h i o

(X c)(X c) = X2 (c + c)X + cc = X2 2aX + (a2 + b2)l a a t h c c o h e s o t h c d a n g (Xa)2 + b2 , n e n v o n g h i e m k h i b = 0, i . e . k h i c R.V i n h a n x e t t r e n v a n h l y p h a n t c h a t h c t r e n t r n g p h c t a c o ( i i ) .

( i i i ) T h e o ( i i ) n e u deg P(X) = n l e , t h P(X) p h a i c o m o t t h a s o (Xc), v i c R,i . e . c o n g h i e m t h c

c.

3 . 8 T m n g h i e m a t h c b a n g p h e p k h a i c a n . P h a n n a y e c a p e n v i e c g i a i

t m n g h i e m a t h c p h c .

P h n g t r n h b a c 2 : ax + bx + c = 0 (a = 0)

C h i a c h o a

:x2 +

b

ax +

c

a= 0

T n h t i e n e k h s o h a n g b a c 1 : X = x+

b

2a, p h n g t r n h c o d a n g

X2 b2 4ac(2a)2

= 0

T m r l a m o t c a n b a c 2 c u a b2 4ac . S u y r a X = r2a

. V a y x =b r

2a

P h n g t r n h b a c 3 : ax + bx + cx + d = 0 (a = 0)

C h i a c h o a: x3 +b

ax2 +

c

ax +

d

a= 0

T n h t i e n e k h s o h a n g b a c 2 : X = x +b

3a, p h n g t r n h c o d a n g X2 +pX+ q = 0

V i e c g i a i p h n g t r n h X2 +pX + q = 0, n h s a u : a t

X = u + v , p h n g t r n h c o d a n g u3 + v3 + (3uv +p)(u + v) + q = 0.T a c a n t m

u, vt h o a h e p h n g t r n h :

u3 + v3 = quv = p

3

a t U = u3, V = v3 . T a c o U + V = q , U V = p3

27.

V a y U, V l a n g h i e m p h n g t r n h X2 + qX p3

27= 0. G i a i t m n g h i e m U0, V0 .

G i a i u3 = U0 , t a c o 3 n g h i e m u0, ju0, j2u0 , v i j = cos(

2

3) + i sin(

2

3) .

G i a i v3 = V0 , t m n g h i e m v0 t h o a p h n g t r n h t h h a i c u a h e u0v0 = p3

.

V a y c o 3 n g h i e m c u a h e l a (u0, v0), (ju0, j2v0), (j

2u0, jv0)


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C h n g 0 . K i e n t h c c h u a n b 1 1

V a y 3 n g h i e m c a n t m : X1 = u0 + v0, X2 = ju0 +j

2v0, X3 = j2u0 +jv0

C a c t n h t o a n t r e n c t o n g k e t b a n g c o n g t h c C a r d a n o :

x = b3a

+q

2+

q2

4+

p3

27+q

2

q2

4+

p3

27

N h a n x e t . T r o n g t h c h a n h c o n g t h c t r e n l a v o d u n g . C h a n g h a n , c o n g t h c t r e n

o i v i x3 21x + 20 = 0 ( c o c a c n g h i e m l a 1, 4,5) :

x =

10 + i

243 +

10 i

243 =???

B a i t a p : G i a i c a c p h n g t r n h ; x3 15x 4 = 0, 2x3 + 18x2 42x + 10 = 0.

P h n g t r n h b a c 4 :

ax + bx + cx + dx + e = 0 (a = 0)C h i a c h o a , r o i t n h t i e n e k h s o h a n g b a c 3 , a t X = x +

b

4a, a v e g i a i p h n g

t r n h :

X4 +pX2 + qX + r = 0

P h a n t c h : X4 +pX2 + qX + r = (X2 + X + )(X2 X + )

o n g n h a t h e s o , t a c o , ,

l a n g h i e m c u a h e :

+ = p + 2

= r

= q

T o 2 l a n g h i e m p h n g t r n h b a c 3 : 2(p + 2)2 4r2 q = 0G i a i t a c o , , . T h a y v a o p h n g t r n h t c h , r o i g i a i p h n g t r n h b a c 2 t a c o c a c

n g h i e m X1, X2, X3, X4

B a i t a p : G i a i p h n g t r n h : x4 + 2x3 + 5x2 + 6x + 9 = 0

P h n g t r n h b a c 5 . A b e l ( 1 8 0 2 - 1 8 2 9 ) a c h n g m i n h k h o n g t h e g i a i m o t p h n g t r n h a t h c b a c 5 t o n g q u a t , t h e o n g h a k h o n g t h e b i e u d i e n n g h i e m n h l a b i e u t h c g o m c a c p h e p t o a n a i s o ( c o n g , t r , n h a n , c h i a ) v a c a n s o ( b a c 2, 3, ) c u a c a c h e s o c u a a t h c . S a u o G a l o i s ( 1 8 1 1 - 1 8 3 2 ) d u n g l y t h u y e t n h o m a t m c

t i e u c h u a n e m o t p h n g t r n h b a c 5 c u t h e c o g i a i c b a n g c a n t h c k h o n g . V d u p h n g t r n h

x5

x 1 = 0 k h o n g g i a i c b a n g c a n t h c

T m n g h i e m h u t c u a p h n g t r n h h e s o n g u y e n . C h o m o t a t h c c o h e s o

n g u y e n

P(X) = a0 + a1X + + anXn ak Z, k = 0, , n , an = 0

K h i o n e u m o t s o h u t

p

q, v i g c d (p,q) = 1, l a n g h i e m c u a P(X) , t h p l a c s o

c u a a0 v a q l a c s o c u a an .


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1 2

C h n g m i n h : N e u

p

ql a n g h i e m c u a

P(X), t h t P(p

q) = 0, t a c o

a0qn + a1q

n1p +

+ an1qp

n1 + anpn = 0

D o g c d (p,q) = 1, t a n g t h c t r e n d e s u y t a p l a c s o c u a a0 , q l a c c u a an .

B a i t a p : T m c a c n g h i e m h u t c u a p h n g t r n h : 3x4 + 5x3 + x2 + 5x 2 = 0

3 . 9 P h a n t h c . M o t p h a n t h c t r e n K l a m o t b i e u t h c d a n g

P(X)

Q(X), t r o n g o P(X), Q(X) K[X], Q(X) = 0

H a i p h a n t h c b a n g n h a u :

P(X)

Q(X)=

P1(X)

Q1(X) P(X)Q1(X) = P1(X)Q(X) .

n h l y . C h oP(X), Q(X) K[X] . G i a s Q(X) = Q1(X)k Qs(X)ks ,

v i Q1(X), , Qs(X) K[X] t h o a i e u k i e n G C D (Qi(X), Qj(X)) = 1, n e u i = j .

K h i o t o n t a i d u y n h a t c a c a t h c A(X), Pij(X) K[X], i = 1, , s , j = 1, , ki ,s a o c h o deg Pij(X) < deg Qi(X) v a

P(X)

Q(X)= A(X) +

si=1

kij=1

Pij(X)

Qi(X)j

C h n g m i n h : S t o n t a i : N e u deg P < deg Q v a Q = D1D2 , v i G C D (D1, D2) = 1 ,t h t h e o a n g t h c B e z o u t t a c o

1 = U1D1 + U2D2, U1, U2 K[X]S u y r a

P = P U1D1 + P U2D2 . C h i a E u c l i d , t a c o P U1 = AD2 + V2, deg V2 < deg D2 .V a y P = V2D1 + V1D2 , v i V1 = AD1 + P U2 . D o deg P < deg Q, deg V2 < deg D2 ,t a c o deg V1 < deg D1 . S u y r a t a c o b i e u d i e n

P

Q=

V1

D1+

V2

D2, v i deg V1 < deg D1, deg V2 < deg D2

T r n g h p t o n g q u a t , t r c h e t c h i a E u c l i d P c h o Q t a c o t h n g l a A. S a u o a p

d u n g b i e u d i e n t r e n c h o D1 = Qk1

, D2 = Qk2 Qkss . T i e p t u c a p d u n g b i e u d i e n

t r e n c h o Q = Qk2

Qkss . S a u h u h a n b c t a c o p h a n t c h

P

Q= A +

si=1

kij=1

Pij

Qji

, v i deg Pij < deg Qi

T n h d u y n h a t : G i a s c o c a c a t h c k h a c A, Pij K[X] , s a o c h o

P

Q= A +

si=1

kij=1

PijQji

, v i deg Pij < deg Qi


16/156

C h n g 0 . K i e n t h c c h u a n b 1 3

S u y r a A = A d o t n h d u y n h a t c u a p h e p c h i a E u c l i d .

T r h a i b i e u d i e n t a c o

si=1

ki

j=1

Pij PijQji

= 0 ().

G i a s p h a n c h n g l a

P1k P1,k = 0. N h a n () v i Q , s u y r a

(P1k P1,k )Qk2 Qkss + Q1U = 0, v i U K[X]

D o G C D (Q1, Qk2 Qkss ) = 1, n e n Q1|(P1k P1,k ) . V a y deg(P1k P1,k ) deg Q1 .

i e u n a y m a u t h u a n v i deg(P1k P1,k ) max(deg P1k , deg P1k ) deg Q1 .V a y p h a i c o P1k P1,k = 0.T n g t l a p l u a n t r e n , t a c o

Pij = Pij , ij .

H e q u a 1 . M o i p h a n t h c t r e n t r n g p h c e u c o p h a n t c h d u y n h a t d i d a n g

P(X)

Q(X) = A(X) +

si=1

mij=1

aij

(X ci)j

t r o n g o A(X) C[X], aij C, c1, , cs C l a c a c n g h i e m c u a Q(X) v i b o i m1, , ms t n g n g .

H e q u a 2 . M o i p h a n t h c t r e n t r n g t h c e u c o p h a n t c h d u y n h a t d i d a n g

P(X)

Q(X)= A(X) +

ri=1

mij=1

aij

(X ci)j +s

i=1

nij=1

bijX + cij(X2 +piX + qi)j

t r o n g o Q(X) = an(X

c1)

m

(X

cr)

ms(X2+p1X+q1)n

(X2+psX+qs)

ns,

X2 +piX + qi (i = 1, , s) k h o n g c o n g h i e m t h c . , A(X) R[X], aij , bij , cij R.

V d u . P h a n t c h

1

X5 X2 t h a n h p h a n t h c n t r e n t r n g t h c :

T a c o X5 X2 = X2(X 1)(X2 + X + 1) .V a y p h a n t c h c o d a n g

1

X5 X2 =A

X+

B

X2+

C

X 1 +DX + E

X2 + X + 1.

e t n h c a c h e s o A , B , C , D , E

d a v a o

1 = AX(X1)(X2+X+1)+B(X1)(X2+X+1)+CX2(X2+X+1)+(DX+E)X2(X1)

o n g n h a t h e s o a t h c 2 v e ( g o i l a p h n g p h a p h e s o b a t n h ) , r o i g i a i h e p h n g

t r n h t u y e n t n h , t a c o

1

X5 X2 =0

X 1

X2+

1

3(X 1) X 1

3(X2 + X + 1)


17/156

I . K h o n g g i a n v e c t o r h n h h o c

C h n g n a y v e c t o r c t r n h b a y d a v a o t r c q u a n , v i m u c c h t a o m o h n h h n h

h o c g i u p c h o v i e c t d u y t r u t n g v a k h a i q u a t h o a c a c c h n g s a u . e c o t h e

l a m v i e c c u t h e h n t r e n c a c v e c t o r , n g u i t a a i s o h o a k h o n g g i a n h n h h o c b a n g

c a c h a v a o h e c s D e s c a r t e s

1

. K h i o c a c p h e p t o a n t r e n v e c t o r s e c o c o n g t h c

t n h t h u a n l i , c o n c a c o i t n g h n h h o c n h n g , m a t c o n g s e c m o t a b i

c a c p h n g t r n h g i u p c h o v i e c n g h i e n c u h n h h o c d e d a n g v a c u t h e h n .

1 . V e c t o r h n h h o c

T r o n g n h i e u v a n e t o a n h o c c u n g n h v a t l y n g o a i c a c a i l n g v o h n g , c o n

c o n h i e u a i l n g c o h n g c h u n g c a c t r n g b i o l n v a h n g , c h a n g h a n

l c , v a n t o c , . . . . C a c a i l n g n a y c m o h n h h o a t h a n h c a c v e c t o r .

1 . 1 n h n g h a . T r o n g k h o n g g i a n E u c l i d E3 m o t v e c t o r c m o t a n h l a m o t

o a n t h a n g c n h h n g AB

.

K y h i e u :

AB= v , h a y n g i a n c h l a v .A g o i i e m g o c , B g o i l a i e m n g o n .

n g t h a n g AB g o i l a p h n g , h n g t A e n B .

o d a i o a n AB

, k y h i e u l a AB .

Q

v

A

B

Q

v

p h n g

H a i v e c t o r

AB ,

CD g o i l a b a n g n h a u

, k y h i e u

AB=

CD , n e u u c h u n g c u n g o d a i v a

h n g , i . e . ABDC l a h n h b n h h a n h .

V e c t o r k h o n g , k y h i e u l a

O h a y O , l a v e c t o r c o g o c t r u n g v i n g o n .

V e c t o r o i c u a

AB c k y h i e u v a n h n g h a AB=

BA .

N h a n x e t . P h a n b i e t n h n g h a t r e n v i k h a i n i e m v e c t o r b u o c k h i t a x e m h a i v e c t o r

c o g o c k h a c n h a u l a k h a c n h a u .

1

R e n e D e s c a r t e s ( 1 5 9 6 - 1 6 5 0 ) v a P i e r r e F e r m a t ( 1 6 0 1 - 1 6 6 5 ) c x e m l a c a c c h a e c u a H n h h o c

g i a i t c h


18/156

1 6

1 . 2 C o n g v e c t o r :

u +

v, c u a h a i v e c t o r

u ,

v, l a v e c t o r x a c n h b i q u i t a c

h n h b n h h a n h h a y q u i t a c h n h t a m g i a c s a u :

0

u

$$$$$$

$$X

v

Q

u +

v

0

u

$$$$$$

$$X v

Q

u +

v

T n h c h a t . V i m o i v e c t o r

u ,

v ,

w, t a c o

u +

v =

v +

u

(

u +

v )+

w =

u +(

v +

w)

v +

O =

v

v +( v ) = O

H e t h c C h a s l e s . T r o n g k h o n g g i a n c h o c a c i e m A0, A1, An . K h i o

A0A1 +

A1A2 + +

An1An =

A0An

1 . 3 N h a n v e c t o r v i s o : v , c u a v e c t o r

vv a s o , l a v e c t o r :

C o o d a i l a

|

|

v

.

C u n g h n g v i

vn e u > 0, n g c h n g v i

vn e u < 0, v a 0

v =

O .

Q

v

t Q

v ( > 0)

C

v ( < 0)

t


u ,

vv a c a c s o , , t a c o

(

v ) = ()

v

( + )

v =

v +

v

(

u +

v ) =

u +

v

1

v =

v


19/156

C h n g I . K h o n g g i a n v e c t o r h n h h o c 1 7

2 . C s D e s c a r t e s - T o a o

e c o t h e l a m v i e c c u t h e h n t r e n c a c v e c t o r , n g i t a a i s o h o a n h s a u

2 . 1 H e c s D e s c a r t e s . D e s c a r t e s a a i s o h o a m a t p h a n g E2 h a y k h o n g g i a n

E u c l i d E3 b a n g c a c h a v a o h e t o a t r u c , m a c h u n g t a a q u e n b i e t v i c a i t e n g o i

h e c s D e s c a r t e s , l a b o b o n (O;

e1,

e2,

e3):

i e m O g o i l a g o c

c u a h e .

C a c v e c t o r

e1,

e2,

e3 , g o i l a c a c v e c t o r c s , t h o a c a c t n h c h a t s a u :

( i )

e1,

e2,

e3 v u o n g g o c v i n h a u t n g o i .

( i i ) e1 = e2 = e3 = 1 .( i i i ) B o b a (

e1,

e2,

e3) t a o t h a n h t a m d i e n t h u a n ( ? ) .

t

OE

e

T

e

e

Ex

y

Tz

tM

C a c n g t h a n g c o v e c t o r c h p h n g e1, e2, e3 t h n g c g o i t e n l a c a c t r u c

Ox,Oy,Ozt n g n g . H e c s t r o n g m a t p h a n g

E2 c n h n g h a t n g t n h

t r o n g E3 . H n n a , t a s e o n g n h a t E2 v i m a t p h a n g Oxy t r o n g E3 , n e n t i e p s a u

a y c a c k h a i n i e m s e c h c t r n h b a y t r o n g k h o n g g i a n .

2 . 2 T o a o . K h i c o n h m o t h e c s D e s c a r t e s (O;

e1,

e2,

e3), t a c o : M o i i e m M E3 t n g n g d u y n h a t m o t b o b a s o (x , y , z), g o i l a t o a o i e m M, x a c n h q u a c a c v e c t o r c s n h p h e p c h i e u v u o n g g o c :

OM= x

e1 +y

e2 +z

e3 .

M o i v e c t o r v , s a u k h i a g o c v e O s e c o n g o n l a A E3 , t n g t n h t r e n

v=

OA=

v1

e1+

v2

e2+

v3

e3 . T a c o t h e m o t a a i s o v e c t o r

vn h b o b a

(v1, v2.v3

)g o i l a

t o a o v e c t o r

v. T a v i e t :

v = (v1, v2, v3) .V i e c v i e t c u n g k y h i e u t o a o c h o v e c t o r v a i e m s e c c h n h r o k h i c a n .

N h a n x e t . N h v a y k h i a v a o h e c s t a a o n g n h a t E3 v i R3

, i . e . t a a

a i s o h o a k h o n g g i a n E3 . C o t h u a n t i e n g ? H a y x e m . . .

2 . 3 M o t a o i t n g h n h h o c b a n g p h n g t r n h h a y p h n g t r n h t h a m s o .

X e t c a c o i t n g h n h h o c ( n g , m a t , k h o i , ) t r o n g m a t p h a n g E2 h a y k h o n g


20/156

1 8

g i a n E3 :

X = {M : M t h o a i e u k i e n (P)}C o n h m o t c s D e s c a r t e s . K h i o c a c i e m M t h a y o i v a t h o a i e u k i e n (P), t h t n g n g c a c t o a o

(x , y , z)c u a

Mc u n g t h a y o i v a t h o a i e u k i e n

(F)n a o o .

C u t h e h n , t a t h n g g a p c a c t r n g h p s a u :

i e u k i e n h n h h o c c o t h e m o t a b i p h n g t r n h :

M t h o a (P) (x , y , z) t h o a p h n g t r n h F(x , y , z) = 0

t r o n g o F : D R, l a m o t h a m s o x a c n h t r e n D R3 .

K h i o t a n o i X

c c h o b i p h n g t r n h F(x , y , z) = 0.

V d u .

a ) T r o n g m a t p h a n g , n g t r o n t a m A = (a, b) b a n k n h R > 0 l a t a p n h n g h a

C = {M E2 : k h o a n g c a c t M e n A b a n g R}

C c c h o b i p h n g t r n h :

F(x, y) = (x a)2 + (y b)2 R2 = 0.b ) T n g t , t r o n g k h o n g g i a n m a t c a u t a m A = (a,b,c), b a n k n h R > 0 c c h o b i p h n g t r n h

F(x , y , z) = (x a)2 + (y b)2 + (z c)2 R2 = 0

T o n g q u a t h n , k h i c a c i e u k i e n (P1), , (Pk) c o t h e m o t a b i c a c p h n g t r n h t n g n g F1 = 0, , Fk = 0, t h

X = {M : M t h o a i e u k i e n (P1) v a v a (Pk)} =k

i=1

{M : M t h o a i e u k i e n (Pi)}

K h i o t a n o i X c c h o b i h e p h n g t r n h : F1(x , y , z) = 0, , Fk(x , y , z) = 0

V d u . H e p h n g t r n h : (x a)2 + (y b)2 + (z c)2 = R2, z = 0,m o t a g i a o c u a m a t c a u v a m a t p h a n g , v a y l a n g t r o n t r e n m a t p h a n g Oxy

i e u k i e n h n h h o c c o t h e m o t a b i p h n g t r n h t h a m s o : K h i t a p a n g x e t l a n g c o n g C ( c h a n g h a n q u y a o c u a m o t i e m ) , n o t h n g c o n

c m o t a b i

M C (x = f(t), y = g(t), z = h(t)), t I

t r o n g o f , g , h : I R l a c a c h a m x a c n h t r e n I R.K h i o t a n o i n g c o n g C c c h o b i p h n g t r n h t h a m s o

x = f(t)y = g(t)z = h(t)

,v i t h a m s o

t I


21/156


T n g t , m a t c o n g S

t h n g c o n c m o t a b i

M S (x = f(s, t), y = g(s, t), z = h(s, t)), (s, t) Dt r o n g o

f , g , h : D

R l a c a c h a m x a c n h t r e n D

R2 .

T a n o i m a t S c c h o b i p h n g t r n h t h a m s o

x = f(s, t)y = g(s, t)z = h(s, t)

, v i 2 t h a m s o (s, t) D

V d u .

a ) n g t r o n t a m A = (a, b), b a n k n h R , c o t h e c h o b i p h n g t r n h t h a m s o t r o n g

m a t p h a n g : x = a + R cos ty = b + R sin t

t [0, 2]

( t t r o n g t r n g h p n a y l a o l n c u a g o c q u a y )

b ) M a t c a u t a m A = (a,b,c), b a n k n h R, c o t h e c h o b i p h n g t r n h t h a m s o t r o n g k h o n g g i a n :

x = a + R cos s sin ty = b + R sin s sin tz = c + R cos t

, (s, t) [0, ] [0, 2]

(s, t

t r o n g t r n g h p n a y c o t h e x e m l a v o v a k i n h o t r e n m a t c a u )

N g o a i r a , c o n n h i e u t r n g h p i e u k i e n h n h h o c c o t h e m o t a b i h e p h n g

t r n h v a b a t p h n g t r n h m a t a k h o n g x e t a y .

3 . C o n g t h c a i s o c u a c a c p h e p t o a n t r e n v e c t o r

T r o n g E3 c o n h h e c s D e s c a r t e s (O;

e1,

e2,

e3).

C h o b a v e c t o r

u = (u1, u2, u3),

v = (v1, v2, v3),

w= (w1, w2, w3).

3 . 1 C o n g t h c c o n g v e c t o r .

u +

v = (u1 + v1, u2 + v2, u3 + v3)

3 . 2 C o n g t h c n h a n v e c t o r v i s o .

v = (v1, v2, v3).

3 . 3 C o n g t h c t n h o d a i . v =

v21

+ v22

+ v23

( c o n g t h c P y t h a g o r e )

3 . 4 C o n g t h c t n h k h o a n g c a c h . C h o c a c i e m A(a1, a2, a3) v a B(b1, b2, b3). K h o a n g

c a c h g i a c h u n g l a o d a i v e c t o r

AB=

OB OA= (b1 a1, b2 a2, b3 a3) . V a y d(A, B) =

AB =

(b1 a1)2 + (b2 a2)2 + b3 a3)2

N g o a i c a c p h e p t o a n c o n g v a n h a n v i s o , c o n c a c p h e p t o a n c b a n t r e n c a c v e c t o r

m a t a s e n h n g h a v a t n h t o a n s a u a y .


22/156

2 0

3 . 5 T c h v o h n g : =

u .

v, c u a h a i v e c t o r

u ,

v, l a m o t s o c n h

n g h a :

= u v cos (u ,v )

V e m a t h n h h o c h n h c h i e u v u o n g g o c c u a

ul e n p h n g

vl a

v 2

v( B a i t a p )

C o n g t h c q u a t o a o :

= u1v1 + u2v2 + u3v3

C h n g m i n h : G o i

u =

OA,

v =

OB . T a c o

AB = v u 2 = u21

+ u22

+ u23

+ v21

+ v22

+ v23

2(u1v1 + u2v2 + u3v3)M a t k h a c , t a c o h e t h c l n g g i a c c h o t a m g i a c OAB :

AB 2 =

OA 2 +

OB 2 2

OA OB cos (AOB)S o s a n h v e p h a i c u a c a c a n g t h c t r e n , t a c o c o n g t h c c a n t m .


u ,

v ,

wv a c a c s o

, , t a c o

=

<

u +

v ,

w> = +

= v 2 0, v a = 0 v = O

3 . 6 T c h h u h n g :

u v =u v , g o i l a t c h v e c t o r c u a h a i v e c t o r u ,v ( t h e o u n g t h t o ) , l a m o t v e c t o r :

u v v u o n g g o c v i m a t p h a n g (u ,v ).(

u ,

v ,

u v ) l a t a m d i e n t h u a n .

u

v =

u

v | sin

(

u ,

v )| =

u 2

v 2

2

V e m a t h n h h o c u v = d i e n t c h h n h b n h h a n h t a o b i u ,v .

E

u

T

u

v

v

e n h n g h a c c h a t c h e t a q u i c

u

vl a v e c t o r

O t r o n g t r n g h p m o t

t r o n g h a i v e c t o r b a n g

O , h a y h a i v e c t o r s o n g s o n g .

V d u .

e1 e1=

O,

e1 e2=e3, e1 e3= e2 .


u v =

e1

e2

e3u1 u2 u3v1 v2 v3

= u2 u3v2 v3

e1 u1 u3v1 v3

e2 + u1 u2v1 v2

e3


23/156


C h n g m i n h : G o i

u v = (x , y , z) . T c a c t n h c h a t

u v u , u v v , u v 2 = u 2 v 2 sin2 (u ,v ) ,t a c o c a c p h n g t r n h t n g n g

u1x + u2y + u3z = 0v1x + v2y + v3z = 0x2 + y2 + z2 = (u1v2 u2v1)2 + (u1v3 u3v1)2 + (u2v3 u3v2)2

G i a i h e p h n g t r n h , k e t h p v i i e u k i e n (

u ,

v ,

u v ) l a t a m d i e n t h u a n , t a c o c o n g t h c t r e n .


u ,

v ,

wv a c a c s o

, , t a c o

u

v =

v

u

(

u +

v )

w =

u

w +

v

w

3 . 7 T c h h o n h p : c u a b a v e c t o r

u ,

v ,

w c n h n g h a l a s o

(

u ,

v ,

w) =V e m a t h n h h o c

|(u,v ,w)| = u v w | cos(u v ,w)|= D i e n t c h a y c h i e u c a o c u a h n h h o p b n h h a n h t a o b i u ,v ,w= T h e t c h h n h h o p b n h h a n h t a o b i

u ,

v ,

w

w

E

u

B

v

T

u

v


(

u,

v ,

w) =

u1 u2 u3v1 v2 v3w1 w2 w3

K h i (u ,v ,w) > 0, i . e . cos(u v ,w) > 0 , h e (u ,v ,w) g o i l a t h u a n .K h i (

u ,

v ,

w) < 0, h e g o i l a n g h c h .

V d u .

a ) H a i v e c t o r

u ,

vv u o n g g o c k h i v a c h k h i

= 0.

b ) H a i v e c t o r

u ,

vs o n g s o n g k h i v a c h k h i

u v =O .c ) B a v e c t o r

u ,

v ,

w o n g p h a n g k h i v a c h k h i (

u ,

v ,

w) = 0 .


24/156

2 2

4 . n g t h a n g v a m a t p h a n g

T r o n g E3 c o n h m o t h e c s D e s c a r t e s . S a u a y l a v a i a p d u n g c u a p h n g

p h a p t o a o .

4 . 1 P h n g t r n h n g t h a n g . n g t h a n g q u a M0 = (x0, y0, z0) c o v e c t o r c h

p h n g

v = (a,b,c) = O c n h n g h a l a t a p

= {M E3 :

M0M s o n g s o n g v i

v }

Q

v

M

M

n g t h a n g

t

Q

V a y M = (x , y , z) n e u v a c h n e u M0M= t v , t R.T o t a c o p h n g t r n h t h a m s o c u a :

x = x0 + tay = y0 + tbz = z0 + tc

, t R

K h i a,b,c = 0, k h t t a c o p h n g t r n h c h n h t a c c u a :x

x0

a =y

y0

b =z

z0

c

4 . 2 P h n g t r n h m a t p h a n g . C h o M0 = (x0, y0, z0) v a h a i v e c t o r k h o n g s o n g s o n g

u = (u1, u2, u3),

v = (v1, v2, v3). M a t p h a n g q u a M0 v a c o c a c v e c t o r c h p h n g

u ,

v

c n h n g h a l a t a p

P = {M E3 :

M0M ,

u ,

v o n g p h a n g }

M Et Mrrrj u

I

v

T

n

P

V a y M = (x , y , z) P n e u v a c h n e u

M0M= s

u +t

v , s , t R.


25/156


T o t a c o p h n g t r n h t h a m s o c u a P

:

x = x0 + su1 + tv2y = y0 + su2 + tv2

z = z0 + su3 + tv3

, s, t R

T a c u n g c o M = (x , y , z) P n e u v a c h n e u (

M0M ,

u ,

v ) == 0.T o t a c o

p h n g t r n h t o n g q u a t c u a P

Ax + By + Cz + D = 0 ,

t r o n g o v e c t o r p h a p

n=

u v = (A , B , C ) = O l a v u o n g g o c v i P.

4 . 3 . M o t s o b a i t o a n . D a v a o t o a o v a p h n g t r n h c o t h e a c a c b a i t o a n

h n h h o c v e b a i t o a n a i s o :

B a i t o a n 1 : X e t v t r c u a 2 m a t p h a n g P1, P2 .

S o g i a o i e m c h n h l a s o n g h i e m c u a h e p h n g t r n h x a c n h b i 2 m a t p h a n g : A1x + B1y + C1z + D1 = 0A2x + B2y + C2z + D2 = 0

P1 P2 A1B2 A2B1, B1C2 B2C1, C1A2 C2A1 k h o n g o n g t h i b a n g 0P1 P2 A1 : A2 = B1 : B2 = C1 : C2 = D1 : D2P1 P2 A1 : A2 = B1 : B2 = C1 : C2 = D1 : D2

B a i t o a n 2 : X e t v t r n g t h a n g v i m a t p h a n g P.T n g t b a i t o a n t r e n x e t h e p h n g t r n h x a c n h b i n g t h a n g v a m a t p h a n g .

x x0

a=

y y0b

=z z0

cAx + By + Cz + D = 0

T a c o M = M0 + t

v P (Aa + Bb + Cc)t + Ax0 + By0 + Cz0 + D = 0. V a y P Aa + Bb + Cc = 0 P Aa + Bb + Cc = 0, Ax0 + By0 + Cz0 + D = 0 P Aa + Bb + Cc = 0, Ax0 + By0 + Cz0 + D = 0

B a i t o a n 3 : T n h g o c g i a c a c m a t p h a n g / n g t h a n g .

V i k y h i e u t r e n t a c o

cos(P1, P2) = | cos(n1, n2)| = |A1A2 + B1B2 + C1C2|A21

+ B21

+ C21

A22

+ B22

+ C22

sin(, P) = | cos(v ,n)| = |Aa + Bb + Cc|A2 + B2 + C2

a2 + b2 + c2

B a i t o a n 4 : T n h k h o a n g c a c h t i e m M0 = (x0, y0, z0) e n m a t p h a n g

P : Ax + By + Cz + D = 0 .


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2 4

c

tM

d

H

T

n

P

G o i H = (x , y , z) l a h n h c h i e u c u a M0 l e n P, t a c o :

M0H n v a H P.T n g n g v i :

M0H= t

nv a Ax + By + Cz + D = 0

T h a y t o a o H v a o p h n g t r n h t r e n : A(x0+ tA) + B(y0+ tB) + C(z0+ tC) + D = 0 .

S u y r a

t =

Ax0 + By0 + Cz0 + D

A2 + B2 + C2V a y

H = (x0, y0, z0) Ax0 + By0 + Cz0 + DA2 + B2 + C2

(A , B , C )

T a c o c o n g t h c t n h k h o a n g c a c h l a

d(M0, P) =

M0H =Ax0 + By0 + Cz0 + D

A2 + B2 + C2

B a i t o a n 5 :

T n h k h o a n g c a c h t i e m M = (x , y , z) e n n g t h a n g q u a M0 v a

c o v e c t o r c h p h n g

v.

EM

v

d

0

M

D a v a o h n h h o c , r o i d u n g c a c p h e p t o a n t r e n v e c t o r , t a c o :

d(M, ) = c h i e u c a o h n h b n h h a n h t a o b i

M0M ,

v

=d i e n t c h h n h b n h h a n h t a o b i

M0M ,

v

v

= M0M v

v


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B a i t o a n 6 : T n h k h o a n g c a c h g i a 2 n g t h a n g 1, 2 .G i a s 1 q u a M1 = (x1, y1, z1) v a c o v e c t o r c h p h n g

v1= (a1, b1, c1)2 q u a M2 = (x2, y2, z2) v a c o v e c t o r c h p h n g

v2= (a2, b2, c2)

s

M

ME

v

B

v

D

D a v a o h n h h o c , r o i d u n g c a c p h e p t o a n t r e n v e c t o r , t a c o :

d(1, 2) = c h i e u c a o h o p b n h h a n h t a o b i

v1,

v2,

M1M2

=t h e t c h h o p b n h h a n h t a o b i

v1,

v2,

M1M2

d i e n t c h h n h b n h h a n h t a o b i

v1,

v2

=|(v1, v2,

M1M2)| v1 v2

B a i t o a n 7 : T m p h n g t r n h n g v u o n g g o c c h u n g c u a 2 n g t h a n g 1, 2 .

V i k y h i e u n h t r e n , g o i A1 1, A2 2 s a o c h o

A1A2 1,

A1A2 1 .K h i o n g v u o n g g o c c h u n g

Dl a n g t h a n g q u a

A1, c o v e c t o r c h p h n g

A1A2.

e x a c n h A1, A2 , t a t i e n h a n h n h s a u :

a t

M1A1= x

v1,

M1A2= y

v2 . S u y r a

A1A2=

M1M2 +y

v2 x v1 .T = 0, = 0, t a c o h e p h n g t r n h :

v1 2x y =

x v2 2y =

G i a i h e p h n g t r n h , t a c o x, y , s u y r a A1, A2 . V a y x a c n h c D .

N h a n x e t . K h i 1 2 v1 v2 v1 v2 = v1 2 v2 2 2= 0:h e p h n g t r n h t r e n d u y n h a t n g h i e m , i . e . c o d u y n h a t m o t n g v u o n g g o c c h u n g .

M o t c a c h l a m k h a c e t m D , d a n h i e u v a o h n h h o c h n : g o i

v =

v1 v2= (a,b,c) ,P1 l a m a t p h a n g q u a M1 v a c o c a c v e c t o r c h p h n g

v1,

v,

P2 l a m a t p h a n g q u a M2 v a c o c a c v e c t o r c h p h n g

v2,

v

K h i o n g v u o n g g o c c h u n g D = P1 P2 . V a y M D k h i v a c h k h i : (

M1M ,

v1,

v ) = 0

(

M2M ,

v2,

v ) = 0


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2 6

T h a y t o a o M = (x , y , z) v a c a c t o a o c u a c a c y e u t o t r e n v a o c a c t c h h o n h p , t a

c o h e p h n g t r n h x a c n h n g v u o n g g o c c h u n g .


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I I . M a t r a n - P h n g p h a p k h G a u s s

T r o n g k h o a h o c , k y t h u a t n h i e u b a i t o a n a v e v i e c t m n g h i e m h e p h n g t r n h t u y e n

t n h . T r o n g c h n g n a y s e e c a p e n p h n g p h a p k h G a u s s

1

, l a p h n g p h a p n

g i a n v a h i e u q u a e g i a i h e p h n g t r n h t u y e n t n h . P h a n a u l a k h a i n i e m v e m a

t r a n

2

, n o l a c o n g c u h u h i e u v a t n h i e n e t h e h i e n , l u t r d l i e u . e s l y c a c s o

l i e u n g i t a a v a o c a c p h e p t o a n t h c h i e n t r e n m a t r a n . N g o a i r a , c h n g n a y

c o n n e u n g d u n g c u a p h e p b i e n o i s c a p t r e n m a t r a n e t n h h a n g , t m m a t r a n

n g c . C a c t n h t o a n c a c c h n g s a u d a n h i e u v a o p h n g p h a p t n h c h n g n a y .

1 . M a t r a n

1 . 1 n h n g h a . M o t m n m a t r a n h a y m a t r a n c a p m n t r e n t r n g s o K( K = R h a y C) l a m o t b a n g c a c s o t h u o c K c s a p x e p n h s a u

A = (aij )mn =

a11 a12 a1na21 a22 a2n

.

.

.

.

.

.

.

.

.

am1 am2 amn

t r o n g o aij K g o i l a p h a n t d o n g i c o t j c u a A.H a i m a t r a n g o i l a b a n g n h a u n e u u c h u n g c o c u n g c a p v a c a c p h a n t c u n g d o n g ,

c u n g c o t b a n g n h a u .

K y h i e u M atK(m, n) l a t a p m o i m n m a t r a n t r e n K.V d u .

a ) M o t m 1 m a t r a n l a m a t r a n c o t . M o t 1 n m a t r a n l a m o t m a t r a n d o n g .

a1a2

.

.

.

am

(x1 x2 xn)

b ) M a t r a n s a u g o i l a m a t r a n l i e n t h u o c c u a o t h b e n c a n h

0 2 0 12 0 2 10 2 0 11 1 1 0

sV1

sV2

sV3

s V4

rrrrrrrr

T r o n g m a t r a n t r e n , p h a n t aij = s o n g n o i Vi v i Vj .1

G o i l a p h n g p h a p k h G a u s s v K a r l F r i e d r i c h G a u s s ( 1 7 7 7 - 1 8 5 5 ) e c a p k h i t n h t o a n e x a c n h

q u y a o h a n h t i n h P a l l a s . T h c r a n g i T r u n g q u o c a b i e t p h n g p h a p n a y t l a u ( k h o a n g 2 0 0 B . C )

2

N g i T r u n g q u o c a s d u n g m a t r a n v a o k h o a n g 2 2 0 0 B . C , x e m v d u v e o v u o n g k y a o c a p 3


30/156

2 8

c ) C a c m a t r a n s a u c g o i l a c a c o v u o n g k y a o

6 1 8

7 5 32 9 4

16 2 3 135 11 10 8

9 7 6 124 14 15 1

15 8 1 24 1716 14 7 5 23

22 20 13 6 43 21 19 12 109 2 25 18 11

C a c m a t r a n t r e n c o c a c p h a n t l a n2 ( n = 3, 4, 5) s o t n h i e n a u t i e n c s a p x e p s a o c h o t o n g c a c s o h a n g t r e n m o i d o n g , m o i c o t , c u n g n h t r e n h a i n g c h e o

c h n h l a m o t h a n g s o !

1 . 2 C a c m a t r a n a c b i e t . C h o A = (aij ) Matk(m, n).M a t r a n k h o n g l a m a t r a n m a m o i p h a n t aij = 0. K y h i e u Omn .M a t r a n v u o n g c a p n l a m a t r a n c a p n n. K y h i e u M atK(n, n) = MatK(n) .M a t r a n n g c h e o l a m a t r a n v u o n g v a c a c p h a n t k h o n g n a m t r e n n g c h e o

e u b a n g

0, i . e .

aij = 0n e u

i = j. K y h i e u

diag(a11, a22, , ann).

M a t r a n n v c a p n l a m a t r a n n g c h e o c a p n m a c a c p h a n t t r e n n g c h e o b a n g 1 , i . e . aij = 0 n e u i = j v a aii = 1. K y h i e u In .M a t r a n t a m g i a c t r e n ( t . . d i ) l a m a t r a n v u o n g m a m o i p h a n t d i ( t . . t r e n )

n g c h e o e u b a n g 0, i . e . aij = 0 n e u i > j ( t . . n e u i < j ) .

diag(1, 2, , n) =

1 0 00 2 0

.

.

.

.

.

.

.

.

.

0 0 n

In =

1 0 0

0 1 0.

.

.

.

.

.

.

.

.

0 0 1

a11 a12 a1n

0 a22 a2n.

.

.

.

.

.

.

.

.

0 0 ann

2 . C a c p h e p t o a n t r e n m a t r a n

2 . 1 C o n g m a t r a n . C h o A = (aij ), B = (bij ) MatK(m, n) .

A + B = (aij + bij ) M atK(m, n)

T n h n g h a p h e p c o n g t a c o :

T n h c h a t . C h o A ,B,C MatK(m, n). K h i o ( i ) A + B = B + A( i i ) (A + B) + C = A + (B + C)( i i i ) A + O = O + A = A , t r o n g o O M atK(m, n) l a m a t r a n k h o n g .

2 . 2 N h a n m o t s o v i m a t r a n . C h o A = (aij ) MatK(m, n) v a K.

A = (aij ) M atK(m, n)


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C h n g I I . M a t r a n - P h n g p h a p k h G a u s s 2 9

T n h n g h a d e k i e m c h n g c a c t n h c h a t s a u :

T n h c h a t . C h o A, B M atK(m, n) v a , K. K h i o ( i ) (A) = ()A( i i ) ( + )A = A + A( i i i ) (A + B) = A + B

V d u .

3

1 2 x0 1 2

2

1 1 y1 0 1

=

1 4 3x 2y

2 3 8

2 . 3 N h a n m a t r a n . C h o A = (aij ) MatK(m, n), B = (bjk ) MatK(n, p) .

AB = (cik) M atK(m, p)

cik = ai1b1k + ai2b2k + + ainbnk =n

j=1

aij bjk

C h u y : C h n h n g h a p h e p n h a n AB k h i s o c o t c u a A = s o d o n g c u a B ( = n) .S o t n h p h a n t cik

ai1 ai2 ain

b1kb2k

.

.

.

bnk

=

cik

V d u .

a )

1 2 34 5 6

0 13 4x y

=

1.0 + 2.3 + 3.x 1.1 + 2.4 + 3.y4.0 + 5.3 + 6.x 4.1 + 5.4 + 6.y

b )

a1 a2 an

x1x2

.

.

.

xn

= a1x1 + a2x2 + + anxn.

c ) C h o

t

ei = (0 1 0)

c o t i

,

A =

a11 a12 a1na21 a22 a2n

.

.

.

.

.

.

.

.

.

.

.

.

am1 a12 amn

, ej =

0.

.

.

1.

.

.

0

d o n g

j.

K h i o

teiA = (ai1 ai2 ain) ( d o n g i c u a A) , Aei =

a1ja2j

.

.

.

amj

( c o t j c u a A)

V a y v i m o i m n m a t r a n A, t a c o : InA = A v a AIm = A.


32/156

3 0

T n h c h a t . C h o A, A, B , B v a C l a c a c m a t r a n . N e u c a c c a c p h e p t o a n c a c b i e u t h c s a u c o n g h a , t h

( i ) A(BC) = (AB)C( i i ) (A + A)B = AB + AB( i i i ) A(B + B) = AB + AB

( i v ) N o i c h u n g AB = BA , i . e . p h e p n h a n m a t r a n k h o n g c o t n h g i a o h o a n .

C h n g m i n h : C a c t n h c h a t ( i i ) , ( i i i ) d e d a n g s u y t n h n g h a . e c h n g m i n h ( i ) , g o i

A = (aij ), B = (bjk ), C = (ckl) . T a c a n s o s a n h A(BC) = (dil) v a (AB)C = (d

il).T h e o n h n g h a , t a c o

dil =

j

aij (

k

bjk ckl) =

j

k

aij bjk ckl

d

il = k

(j

aij bjk )ckl = k

j

aij bjk ckl

V a y dij = d

ij , i, j .

D e t m v d u e c h n g m i n h ( i v ) , c h a n g h a n A =

0 10 0

v a B =

0 00 1

.

K h i o AB = A c o n BA = O .

2 . 4 M a t r a n n g c . M o t m a t r a n v u o n g A M atK(n) g o i l a k h a n g h c h n e u u t o n t a i X MatK(n) s a o c h o XA = AX = In .K h i o d e t h a y X l a d u y n h a t , g o i l a m a t r a n n g c c u a A v a k y h i e u l a A1 .V a y t h e o n h n g h a , t a c o A1A = AA1 = InK y h i e u GlK(n) l a t a p m o i m a t r a n v u o n g c a p n t r e n K k h a n g h c h .

T n h c h a t . C h o A, B GlK(n) . K h i o A1, B1, AB GlK(n) v a

( i ) (A1)1 = A. ( i i ) (AB)1 = B1A1 .

C h n g m i n h : D o AA1 = A1A = In , n e n A1

c o m a t r a n n g c l a A.D o (AB)(B1A1) = A(BB1)A1 = AIA1 = AA1 = I.T n g t , (B1A1)AB = I. V a y t a c o ( i i ) .

V d u .

a ) e t n h m a t r a n n g c c u a A = 2 13 2 t a g i a i p h n g t r n h AX = I2 v i a n X MatK(2) . S u y r a m a t r a n n g c l a A

1 =

2 1

3 2

.

b ) D e t m v d u m a t r a n k h o n g k h a n g h c h , c h a n g h a n m a t r a n k h o n g , h a y

1 12 2

V i e c x a c n h m o t m a t r a n k h a n g h c h k h i n a o v a t n h m a t r a n n g c s e c e

c a p s a u .


33/156


2 . 5 P h e p c h u y e n v m a t r a n . C h o A = (aij ) M atK(m, n). M a t r a n c h u y e n v c u a m a t r a n A c k y h i e u v a n h n g h a n h s a u

t

A = (t

aji ) M atK(n, m) ,t r o n g o

t

aji = aiji . e . c h u y e n d o n g c u a A t h a n h c o t c u a tA.

V d u . M a t r a n c h u y e n v c u a m a t r a n

1 2 34 5 6

l a

1 42 5

3 6

T n h c h a t . C h o A ,B,C ,D l a c a c m a t r a n . N e u c a c p h e p t o a n t r o n g c a c b i e u t h c s a u c o n g h a , t h

( i )

t(A + B) = tA + tB( i i )

t(A) = tA( i i i )

t

(AC) =t

Ct

A( i v )

t(D1) = (tD)1

C h n g m i n h : ( i ) , ( i i ) v a ( i i i ) d e k i e m c h n g . e c h n g m i n h ( i v ) , g a s D GlK(n) .T ( i i i ) s u y r a

tD t(D1) = t(D1D) = tIn = In . T n g t t(D1) tD = In .

V a y

t(D1) l a m a t r a n n g c c u a tD .

2 . 6 L u y t h a m a t r a n . C h o A l a m o t m a t r a n v u o n g c a p n . V i m o i k N, n h n g h a

A0 = In, A1 = A, Ak = AA A

k l a n

V d u . B a n g q u i n a p c o t h e c h n g m i n h v i m o i k N,cos sin sin cos

k=

cos k sin ksin k cos k

diag(1, 2, , n)k = diag(k1,

k2, ,

kn)

2 . 7 B i e n o i s c a p t r e n m a t r a n . C a c p h e p b i e n o i t r e n m a t r a n s a u a y c g o i

l a b i e n o i s c a p

:

B i e n o i 1 - C h u y e n v h a i d o n g ( c o t ) .

B i e n o i 2 - N h a n m o t d o n g ( c o t ) v i m o t s o k h a c k h o n g .

B i e n o i 3 - C o n g m o t d o n g ( c o t ) v i b o i c u a m o t d o n g ( c o t ) k h a c .

K y h i e u .

Di Dj P h e p c h y e n v d o n g t h i v a t h j.Di P h e p n h a n d o n g t h i v i s o .Di + Dj P h e p c o n g d o n g t h i v i l a n d o n g t h j.Ci Cj P h e p c h y e n v c o t t h i v a t h j.Ci P h e p n h a n c o t t h i v i s o .Ci + Cj P h e p c o n g c o t t h i v i l a n c o t t h j.


34/156

3 2

C h o A, B MatK(m, n). K h i o B g o i l a t n g n g s c a p v i A. , k y h i e u A B ,n e u u B n h a n c t A q u a h u h a n b i e n o i s c a p t r e n A.

N h a n x e t . C h o A M atK(m, n), B M atK(n, p).K y h i e u ai l a d o n g t h i c u a A, bj l a c o t t h j c u a BT h e o q u i t a c n h a n m a t r a n t a c o

AB =

a1

a2

.

.

.

am

(b1 b2 bp). =

a1Ba2B

.

.

.

amB

= (Ab1 Ab2 Abp) ,

i . e . d o n g i c u a AB = ( d o n g i c u a A) B , v a c o t j c u a AB = A ( c o t j c u a B ) .V a y b i e n o i s c a p t r e n d o n g AB = ( b i e n o i s c a p t r e n d o n g A) B ,v a b i e n o i s c a p t r e n c o t AB = A( b i e n o i s c a p t r e n c o t B ) .S u y r a c a c b i e n o i s c a p c o t h e b i e u d i e n q u a p h e p n h a n m a t r a n :

M e n h e . C h o A MatK(m, n). K h i o ( i ) N e u d l a b i e n o i s c a p t r e n d o n g c u a A, t h d(A) = d(Im)A ( n h a n b e n t r a i ) . ( i i ) N e u c l a b i e n o i s c a p t r e n c o t c u a A, t h c(A) = Ac(In) ( n h a n b e n p h a i ) .

M a t r a n s c a p l a m a t r a n n h a n t b i e n o i s c a p m a t r a n n v .

B a i t a p : H a y l i e t k e m o i d a n g c u a m a t r a n s c a p c a p 2 : d(I2), c(I2).C h n g m i n h c a c m a t r a n s c a p l a k h a n g h c h . T m c a c m a t r a n n g c t n g n g .

V d u .

a ) 0 1 01 0 0

0 0 1

1 a b c2 d e f

3 g h i

=

2 d e f1 a b c

3 g h i

(d = D1 D2)

1 2 3x y z

t u v

1 0 0 1 0

0 0 1

=

1 2 3 + x y z + x

t u v + t

(c = C3 + C1)

b ) G a s a11 = 0 . C h o

A =

a11 a12 a1na21 a22 a2n

.

.

.

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.

.

.

.

.

.

am1 a12 amn

va E =

1 0 0 0

aa 1 0 0 a

a0 1 0

.

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.

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.

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.

.

ama

0 0 1

B a i t a p : M a t r a n E b i e u d i e n p h e p b i e n o i g t r e n d o n g c u a m a t r a n n v Im ?V i e t c o t a u c u a m a t r a n EA .


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2 . 8 M a t r a n d a n g b a c t h a n g . C h o A = (aij ) M atK(m, n) . K h i o A g o i l a c o d a n g b a c t h a n g n e u u

aij = 0, j < j aij = 0, i

> i,

i . e . n e u t r e n d o n g i c a c p h a n t p h a t r a i aij e u b a n g 0, t h t r e n c o t j m o i p h a n t p h a d i aij e u b a n g 0 .

n e u

0 0 0 aij

00

.

.

.

0

t h

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0

( = 0)

2 . 9 M e n h e . M o i m a t r a n e u c o t h e a v e d a n g b a c t h a n g b a n g h u h a n p h e p b i e n

o i s c a p t r e n d o n g .

V i e c c h n g m i n h c t h e h i e n q u a t h u a t t o a n s a u :

T h u a t t o a n G a u s s .

I n p u t : A M atK(m, n)

O u t p u t : B M atk(m.n) d a n g b a c t h a n g v a A BT h u a t t o a n c t i e n h a n h q u i n a p ( l a p ) n h s a u :

G a s v o n g l a p t h k 1, k 1 d o n g a u c u a A c o d a n g b a c t h a n g . B a y g i c h t h c h i e n b i e n o i t r e n c a c d o n g i k .T r n g h p 1 : M o i p h a n t m o i d o n g i k b a n g 0 . K h i o A a c o d a n g b a c t h a n g . T r n g h p 2 : T m p h a n t c o v t r k h a c 0 g a n p h a t r a i n h a t

j(k) = min{j : aij = 0, i k}

B c 1 : C h u y e n v d o n g , Di Dk , e d o n g k c o akj(k) = 0 .B c 2 : K h c a c p h a n t c o t j(k) c a c d o n g i > k b a n g b i e n o i :

Di iDk, v i i =aij(k)

akj(k)

B c 3 : N e u k < m, t a n g k l e n 1.T i e n h a n h l a p n h t r e n , s a u h u h a n ( m) b c , t a c o m a t r a n B d a n g b a c t h a n g .

H a n g c u a m a t r a n A l a s o n g u y e n r , k y h i e u r = rankA, n e u u c o t h e d u n g c a c p h e p b i e n o i s c a p a A v e d a n g b a c t h a n g v i r d o n g k h a c k h o n g . V i e c c h n g m i n h t n h u n g a n c u a n h n g h a v a n e u : h a i m a t r a n t n g n g s

c a p l a c o c u n g h a n g ( i . e . r k h o n g p h u t h u o c b i e n o i s c a p ) c c h n g m i n h c a c


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3 4

c h n g s a u . a y c h n e u a p d u n g c u a t h u a t t o a n t r e n e t n h h a n g .

S o t n h h a n g :

A B ( d a n g b a c t h a n g ) , rankA = s o d o n g k h a c k h o n g c u a B

V d u . C h o

A =

1 0 2 21 1 3 2

2 1 5 0

e t n h h a n g c u a A t a d u n g c a c p h e p b i e n o i t r e n d o n g m a t r a n n h s a u

1 0 2 21 1 3 22 1 5 0

D D ,D 2D

1 0 2 20 1 1 40 1 1 4

D3D2

1 0 2 20 1 1 40 0 0 0

T o s u y r a rankA = 2.

e k e t t h u c t i e t n a y t a p h a t b i e u n h l y s a u , v i e c c h n g m i n h x e m n h b a i t a p .

2 . 1 0 n h l y . C h o A M atK(m, n) . K h i o rankA = r k h i v a c h k h i t o n t a i

h u h a n p h e p b i e n o i s c a p t r e n d o n g v a t r e n c o t e A

Ir 00 0

. .

N o i m o t c a c h k h a c , t o n t a i c a c m a t r a n k h a n g h c h P GlK(m), Q GlK(n), s a o c h o

P AQ =

1 0 0.

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.

0 1 00 0 0

.

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0 0 0

r

H n g d a n : D u n g t h u a t t o a n G a u s s b i e n o i s c a p t r e n d o n g ( = n h a n b e n t r a i A v i c a c m a t r a n s c a p ) a A v e d a n g b a c t h a n g . T n g t , c o t h e d u n g c a c b i e n o i s c a p t r e n c o t ( = n h a n p h a i b i c a c m a t r a n s c a p ) c u a m o t m a t r a n d a n g b a c t h a n g

a m a t r a n o v e d a n g n g c h e o n h t r e n . G o i

P( t . .

Q) l a t c h c a c m a t r a n s

c a p t n g n g v i p h e p b i e n o i t r e n d o n g ( t . . c o t ) e y l a c a c m a t r a n s c a p l a

k h a n g h c h , n e n P, Q k h a n g h c h .

B a i t a p : T h c h i e n c u t h e h n g d a n t r e n t m P, Q c h o m a t r a n v d u p h a n t r e n . T o v i e t t h u a t t o a n c h o n h l y t r e n .


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3 . P h n g p h a p k h G a u s s

3 . 1 n h n g h a . M o t h e p h n g t r n h t u y e n t n h m p h n g t r n h , n a n , t r e n t r n g K, l a b i e u t h c d a n g

a11x1 + a12x2 + + a1nxn = b1a21x1 + a22x2 + + a2nxn = b2

.

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.

am1x1 + am2x2 + + amnxn = bm

t r o n g o aij , bj K, v a x1, , xn l a c a c k y h i e u g o i l a c a c a n .N e u b1 = = bn = 0, t h h e g o i l a h e p h n g t r n h t h u a n n h a t . B o n s o (x1, , xn) K

n, t h o a c a c p h n g t r n h t r e n g o i l a n g h i e m c u a h e .

H e p h n g t r n h g o i l a t n g t h c h n e u u n o c o t a p n g h i e m k h a c t r o n g .

B a i t o a n .

1 - K h i n a o h e t n g t h c h , i . e . c o n g h i e m ?

2 - K h i h e c o n g h i e m t h d u y n h a t n g h i e m h a y b a o n h i e u n g h i e m ?

3 - G i a i h e , i . e . t m t a p n g h i e m c u a h e .

3 . 2 B i e u d i e n m a t r a n h e p h n g t r n h t u y e n t n h . N e u k y h i e u

A =

a11 a12 a1na21 a22 a2n

.

.

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.

.

.

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.

.

.

.

am1 a12 amn

, x =

x1x2

.

.

.

xn

, b =

b1b2

.

.

.

bm

,

t h t p h e p n h a n m a t r a n h e p h n g t r n h c o t h e v i e t g o n t h a n h Ax = b.

N h a n x e t . P h n g p h a p t h e l a p h n g p h a p n g i a n n h a t e g i a i h e p h n g t r n h

t u y e n t n h . G i a s h e c o h e s o a11 = 0. K h i o t p h n g t r n h a u

x1 =1

a11( b1 a12x2 a1nxn ).

T h e b i e u t h c t r e n v a o c a c p h n g t r n h c o n l a i , t a a v e v i e c g i a i m 1 p h n g t r n h , n 1 a n x2, , xn . T i e p t u c t i e n h a n h t n g t c h o h e m i . . . .

e g i a i h e p h n g t r n h c l n , n g i t a t h n g d u n g p h n g p h a p k h G a u s s ( h a y

c a i t i e n c u a p h n g p h a p n a y )

3

. P h n g p h a p n a y d a v a o n h a n x e t n g i a n s a u :

3 . 3 M e n h e . C a c p h e p b i e n o i s c a p s a u a y a h e p h n g t r n h v e h e t n g

n g v i h e a c h o , i . e . h e a b i e n o i c o c u n g t a p n g h i e m v i h e x u a t p h a t :

B i e n o i 1 - C h u y e n v h a i p h n g t r n h c u a h e .

B i e n o i 2 - N h a n m o t p h n g t r n h v i m o t s o k h a c k h o n g .

B i e n o i 3 - C o n g m o t p h n g t r n h v i b o i m o t p h n g t r n h k h a c c u a h e .

3

X e m c h n g n h t h c e b i e t a n h g i a s o p h e p t o a n c a n e t h c h i e n t h u a t t o a n G a u s s


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3 6

N h a n x e t . C a c p h e p b i e n o i s c a p t r e n t n g n g v i b i e n o i s c a p t r e n d o n g m a

t r a n m r o n g (A | b) c u a h e .

3 . 4 n h l y . M o i h e p h n g t r n h t u y e n t n h , b a n g c a c p h e p b i e n o i s c a p , e u

t n g n g v i h e c o d a n g b a c t h a n g . i . e . h e c o d a n g Ax = b v i A l a m a t r a n d a n g b a c t h a n g .

V i e c c h n g m i n h n h l y t r e n c t h e h i e n q u a p h n g p h a p s a u :

3 . 5 P h n g p h a p k h G a u s s .

G i a i h e Ax = b , t r o n g o A = (aij ) M atK(m, n), b Km

.

S o c u a p h n g p h a p :

B c 1 : D u n g t h u a t t o a n G a u s s (A | b) (A | b) (A c o d a n g b a c t h a n g )B c 2 : G i a i p h n g t r n h Ax = b b a n g p h n g p h a p t h e .

N h a n x e t . R a t d e g i a i h e p h n g t r n h d a n g b a c t h a n g b a n g p h n g p h a p t h e . C h a n g

h a n , e g i a i h e p h n g t r n h s a u :

2x1 + x2 + x3 3x4 + x5 = 13x2 + x3 x4 + x5 = 2

x4 + x5 = 3

B a t a u t p h n g t r n h c u o i , l a n l t t a c o :

x4 = 3 x5, x2 = 1/3(2 + ), x1 = 1/2(1 + )

V a y h e t r e n c o v o s o n g h i e m , p h u t h u o c 2 t h a m s o x3 v a x5 .

3 . 6 B i e n l u a n . G a s h e Ax = b c o d a n g b a c t h a n g , i . e .

(A | b) =

a1j(1) a1j(2) a1j(r) a1n | b10 a2j(2) a2j(r) a2n | b2

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. |.

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.

0 0 arj(r) arn | br0 0 0 0 | br+1

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.

. |.

.

.

0 0 0 0 | bm

T r o n g o m l a s o p h n g t r n h , n l a s o a n , v a r c h n h l a h a n g c u a A.T r n g h p 1 : r = m = n , h e d u y n h a t n g h i e m . T r n g h p 2 : r = m < n, h e c o v o s o n g h i e m . C a c b i e n xj(1), , xj(r) c g i a i t h e o n r b i e n c o n l a i , g o i l a c a c b i e n t d o c o n s o (n r) g o i l a b a c t d o .T r n g h p 3 : r < m,

3 a ) br+1 = = bm = 0, h e c o n g h i e m ( v i b a c t d o l a n r ) .3 b ) (br+1, , bm) = (0, , 0), h e v o n g h i e m .


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M i n h h o a t r n g h p 1 :

| 0 | 0 0 | 0 0 0 | 0 0 0 0 |

M i n h h o a t r n g h p 2 :

| 0 | 0 0 0 | 0 0 0 0 |

M i n h h o a t r n g h p 3 a v a 3 b :

| | 0 0 | | 0 0 0 | |

0 0 0 0 0 | | 0 0 0 0 0 0 | 0 | 0 0 0 0 0 0 | 0 |

V d u . G i a i o n g t h i 2 h e p h n g t r n h c h k h a c n h a u c o t b

x2 + x3 3x4 + 2x5 = 6 | 32x1 x2 + 2x4 6x5 = 4 | 66x1 3x2 + 7x4 10x5 = 8 | 102x1 + x3 + x4 + 12x5 = 15 | 7

T a t h c h i e n c a c p h e p b i e n o i t r e n m a t r a n m r o n g :

2 1 0 2 6 | 4 | 60 1 1 3 2 | 6 | 36 3 0 7 10 | 8 | 102 0 1 1 12 | 15 | 7

(D1 D2)

1 1/2 0 1 3 | 2 | 30 1 1 3 2 | 6 | 30 0 0 1 8 | 20 | 80 1 1 1 18 | 11 | 13

(1/2D1, D3 3D1, D4 2D1)

1 1/2 0 1 3 | 2 | 30 1 1 3 2 | 6 | 30 0 0 1 8 | 20 | 80 0 0 2 16 | 5 | 16

(D4 D2)

1 1/2 0 1 3 | 2 | 30 1 1 3 2 | 6 | 30 0 0 1 8 | 20 | 80 0 0 0 0 | 45 | 0

(D4 2D3)


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3 8

S a u b i e n o i h e c o d a n g b a c t h a n g

x1 1/2x2 + x4 3x5 = 2 | 3x2 + x3 3x4 2x5 = 6 | 3

x4 + 8x5 = 20 | 80 = 45 | 0

V a y h e v i c o t a u v o n g h i e m v c o p h t r n h c u o i 0 = 45 .H e v i c o t s a u c o p h n g t r n h c u o i l a t h a (0 = 0) . T i e p t u c g i a i b a n g p h n g p h a p t h e t p h n g t r n h c u o i c u a h e , t a c o

x4 = 8 8x5x2 = 3 x3 + 3x4 x5 = 21 x3 26x5x1 = 8 1/2x2 x4 + 3x5 = 1/2 1/2x3 2x5

N g h i e m t o n g q u a t c u a h e v i c o t s a u :

(1/2 1/2x3 2x5, 21 x3 26x5, x3, 8 8x5, x5), x3, x5 K.

V a y h e c o v o s o n g h i e m p h u t h u o c 2 t h a m s o , i . e . h e c o b a c t d o l a 2 .

N h a n x e t . C o t h e g i a i t i e p h e p h n g t r n h d a n g b a c t h a n g b a n g b i e n o i s c a p

t r e n m a t r a n . T n g t n h p h a n c u o i c u a t h u a t t o a n s a u :

3 . 7 P h n g p h a p G a u s s - J o r d a n t n h m a t r a n n g c . C h o A M atK(n).N h a n x e t . S t o n t a i v a v i e c t m m a t r a n n g c c u a A t n g n g v i v i e c g i a i o n g t h i n h e p h n g t r n h n a n . C u t h e , g o i Xi , i = 1, , n, l a n g h i e m ( n e u c o ) c u a h e p h n g

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