Bài tập toán cao cấp Tập 1 part 8 pptx

5.2. Co
.
so
.
’
.D
-
ˆo
’
ico
.
so
.
’
195
1
+
Ch´u
.
ng minh r˘a
`
ng E
1
,E
2
lˆa
.
p th`anh co
.
so
.

’
cu
’
a R
2
.
2
+
T`ım to
.
adˆo
.
vecto
.
x trong co
.
so
.
’
E
1
,E
2
.
3
+
T`ım to
.
adˆo
.

cu
’
a vecto
.
x trong co
.
so
.
’
E
2
,E
1
.
Gia
’
i. 1
+
Ta lˆa
.
p ma trˆa
.
n c´ac to
.
adˆo
.
cu
’
a E
1

v`a E
2
:
A =

1 −2
21

⇒ detA =5=0.
Do d
´ohˆe
.
hai vecto
.
E
1
,E
2
l`a dltt trong khˆong gian 2-chiˆe
`
u R
2
nˆen n´o
lˆa
.
p th`anh co
.
so
.
’

.
2
+
Trong co
.
so
.
’
d˜a cho vecto
.
x c´o to
.
adˆo
.
l`a (3, −4). Gia
’
su
.
’
trong
co
.
so
.
’
E
1
,E
2
vecto

.
x c´o to
.
adˆo
.
(x
1
,x
2
). Ta lˆa
.
p ma trˆa
.
n chuyˆe
’
nt`u
.
co
.
so
.
’
E
1
, E
2
dˆe
´
nco
.

so
.
’
E
1
,E
2
:
T =

12
−21

⇒ T
−1
=
1
5

12
−21

Khi d
´o

x
1
x
2


= T
−1

3
−4

⇒

x
1
x
2

=
1
5

1 −2
21

3
−4

=
1
5

11
2


=



11
5
2
5



.
Vˆa
.
y x
1
=
11
5
, x
2
=
+2
5
.
3
+
V`ı E
1
,E

2
l`a co
.
so
.
’
cu
’
a R
2
nˆen E
2
,E
1
c˜ung l`a co
.
so
.
’
cu
’
a R
2
.Ma
trˆa
.
n chuyˆe
’
nt`u
.

co
.
so
.
’
E
1
, E
2
dˆe
´
nco
.
so
.
’
E
2
,E
1
c´o da
.
ng
A
∗
=

21
1 −2


,A
∗
−1
= −
1
5

−2 −1
−12

3
−4

= −
1
5

−2
−11

=



2
5
11
5




Do d
´o x
1
=
2
5
, x
2
=
11
5
trong co
.
so
.
’
E
2
,E
1
.
V´ı d u
.
8. Trong khˆong gian R
3
cho co
.
so
.

’
E
1
, E
2
, E
3
n`ao d´o v`a trong
co
.
so
.
’
d´o c´ac vecto
.
E
1
,E
2
,E
3
v`a x c´o to
.
adˆo
.
l`a E
1
=(1, 1, 1); E
2
=

(1, 2, 2), E
3
=(1, 1, 3) v`a x =(6, 9, 14).
196 Chu
.
o
.
ng 5. Khˆong gian Euclide R
n
1
+
Ch´u
.
ng minh r˘a
`
ng E
1
,E
2
,E
3
c˜ung lˆa
.
p th`anh co
.
so
.
’
trong R
3

.
2
+
T`ım to
.
adˆo
.
cu
’
a x trong co
.
so
.
’
E
1
,E
2
,E
3
.
Gia
’
i. 1
+
tu
.
o
.
ng tu

.
.
nhu
.
trong v´ı du
.
7, ha
.
ng cu
’
ahˆe
.
ba vecto
.
E
1
,E
2
,E
3
b˘a
`
ng 3 nˆen hˆe
.
vecto
.
d
´odˆo
.
clˆa

.
p tuyˆe
´
n t´ınh trong khˆong
gian 3-chiˆe
`
u nˆen n´o lˆa
.
p th`anh co
.
so
.
’
cu
’
a R
3
.
2+ Dˆe
’
t`ım to
.
adˆo
.
cu
’
a x trong co
.
so
.

’
E
1
,E
2
,E
3
ta c´o thˆe
’
tiˆe
´
n h`anh
theo hai phu
.
o
.
ng ph´ap sau.
(I) V`ı E
1
,E
2
,E
3
lˆa
.
p th`anh co
.
so
.
’

cu
’
a R
3
nˆen
x = x
1
E
1
+ x
2
E
2
+ x
3
E
3
⇒ (6, 9, 14) = x
1
(1, 1, 1) + x
2
(1, 2, 2) + x
3
(1, 1, 3)
v`a do d´o x
1
,x
2
,x
3

l`a nghiˆe
.
mcu
’
ahˆe
.
x
1
+ x
2
+ x
3
=6,
x
1
+2x + x
3
=9,
x
1
+2x
2
+3x
3
=14.






⇒ x
1
=
1
2
,x
2
=3,x
3
=
5
2
·
(I I) Lˆa
.
p ma trˆa
.
n chuyˆe
’
nt`u
.
co
.
so
.
’
E
1
, E
2

, E
3
sang co
.
so
.
’
E
1
,E
2
,E
3
:
T
EE
=



111
121
123



⇒ T
−1
EE
=

1
2



4 −1 −1
−22 0
0 −11



.
Do d´o



x
1
x
2
x
3



= T
−1
EE




6
9
14



=
1
2



1
6
5



=





1
2
3
5
2






v`a thu du
.
o
.
.
ckˆe
´
t qua
’
nhu
.
tronng (I). 
B
`
AI T
ˆ
A
.
P
5.2. Co
.
so
.
’
.D

-
ˆo
’
ico
.
so
.
’
197
1. Ch´u
.
ng minh r˘a
`
ng c´ac hˆe
.
vecto
.
sau dˆay l`a nh˜u
.
ng co
.
so
.
’
trong khˆong
gian R
4
:
1) e
1

=(1, 0, 0, 0); e
2
=(0, 1, 0, 0); e
3
=(0, 0, 1, 0); e
4
=(0, 0, 0, 1).
2) E
1
=(1, 1, 1, 1); E
2
=(0, 1, 1, 1); E
3
=(0, 0, 1, 1); E
4
=(0, 0, 0, 1).
2. Ch´u
.
ng minh r˘a
`
ng hˆe
.
vecto
.
do
.
nvi
.
:
e

1
=(1, 0, ,0
  
n−1
); e
2
=(0, 1, 0, ,0), ,e
n
=(0, 0, ,0
  
n−1
, 1)
lˆa
.
p th`anh co
.
so
.
’
trong R
n
.Co
.
so
.
’
n`ay du
.
o
.

.
cgo
.
il`aco
.
so
.
’
ch´ınh t˘a
´
c.
3. Ch´u
.
ng minh r˘a
`
ng hˆe
.
vecto
.
E
1
=(1, 0, ,0),
E
2
=(1, 1, ,0),

E
n
=(1, 1, ,1)
l`a mˆo

.
tco
.
so
.
’
trong R
n
.
4. Ch´u
.
ng minh r˘a
`
ng hˆe
.
vecto
.
E
1
=(1, 2, 3, ,n− 1,n),
E
2
=(1, 2, 3, ,n− 1, 0),

E
n
=(1, 0, 0, ,0, 0)
lˆa
.
p th`anh co

.
so
.
’
trong khˆong gian R
n
.
5. H˜ay kiˆe
’
m tra xem mˆo
˜
ihˆe
.
vecto
.
sau dˆa y c ´o l ˆa
.
p th`anh co
.
so
.
’
trong
khˆong gian R
4
khˆong v`a t`ım c´ac to
.
adˆo
.
cu

’
a vecto
.
x =(1, 2, 3, 4) trong
mˆo
˜
ico
.
so
.
’
d´o.
1) a
1
=(0, 1, 0, 1); a
2
=(0, 1, 0, −1); a
3
=(1, 0, 1, 0);
a
4
=(1, 0, −1, 0). (DS. 3, −1, 2, −1)
2) a
1
=(1, 2, 3, 0); a
2
=(1, 2, 0, 3); a
3
=(1, 0, 2, 3);
198 Chu

.
o
.
ng 5. Khˆong gian Euclide R
n
a
4
=(0, 1, 2, 3). (DS.
2
3
, −
1
6
,
1
2
, 1)
3) a
1
=(1, 1, 1, 1); a
2
=(1, −1, 1, −1); a
3
=(1, −1, 1, 1);
a
4
=(1, −1, −1, −1). (DS.
3
2
, −

1
2
, 1, −1)
4) a
1
=(1, −2, 3, −4); a
2
=(−4, 1, −2, 3); a
3
=(3, −4, 1, −2);
a
4
=(−2, 3, −4, 1). (DS. −
13
10
, −
7
10
, −
13
10
, −
17
10
)
Nhˆa
.
nx´et. Ta nh˘a
´
cla

.
ir˘a
`
ng c´ac k´y hiˆe
.
u e
1
,e
2
, ,e
n
du
.
o
.
.
cd`ung dˆe
’
chı
’
c´ac vecto
.
do
.
nvi
.
cu
’
a tru
.

c x
i
(i =1, 2, ,n):
e
i
=(1, 0, ,0

 
n−1
),e
2
=(0, 1, 0, ,0), ,e
n
=(0, ,0

 
n−1
, 1)
6. T`ım ma trˆa
.
n chuyˆe
’
nt`u
.
co
.
so
.
’
e

1
,e
2
,e
3
dˆe
´
nco
.
so
.
’
e
2
,e
3
,e
1
.
(D
S.



001
100
010




)
7. T`ım ma trˆa
.
n chuyˆe
’
nt`u
.
co
.
so
.
’
e
1
,e
2
,e
3
,e
4
dˆe
´
nco
.
so
.
’
e
3
,e

4
,e
2
,e
1
.
(D
S.





0001
0010
1000
0100





)
8. Cho ma trˆa
.
n

−11
20


l`a ma trˆa
.
n chuyˆe
’
nt`u
.
co
.
so
.
’
e
1
,e
2
dˆe
´
nco
.
so
.
’
E
1
, E
2
.T`ım to
.
adˆo
.

cu
’
a vecto
.
E
1
, E
2
.
(DS. E
1
=(−1, 2); E
2
=(1, 0))
9. Gia
’
su
.
’



12−1
31 0
20 1



5.2. Co
.

so
.
’
.D
-
ˆo
’
ico
.
so
.
’
199
l`a ma trˆa
.
n chuyˆe
’
nt`u
.
co
.
so
.
’
e
1
,e
2
,e
3

dˆe
´
nco
.
so
.
’
E
1
, E
2
, E
3
.T`ım to
.
adˆo
.
cu
’
a vecto
.
E
2
trong co
.
so
.
’
e
1

,e
2
,e
3
.(DS. E
2
=(2, 1, 0))
10. T`ım ma trˆa
.
n chuyˆe
’
nt`u
.
co
.
so
.
’
e
1
,e
2
,e
3
dˆe
´
nco
.
so
.

’
E
1
=2e
1
− e
3
+ e
2
; E
2
=3e
1
− e
2
+ e
3
; E
3
= e
3
.
(D
S.



230
1 −10
−111




)
11. T`ım ma trˆa
.
n chuyˆe
’
nt`u
.
co
.
so
.
’
e
1
,e
2
,e
3
dˆe
´
nco
.
so
.
’
E
1

= e
2
+ e
3
; E
2
= −e
1
+2e
3
; E
3
= e
1
+ e
2
.
(DS.



0 −11
101
120



)
12. T`ım ma trˆa
.

n chuyˆe
’
nt`u
.
co
.
so
.
’
e
1
,e
2
,e
3
,e
4
dˆe
´
nco
.
so
.
’
E
1
=2e
2
+3e
3

+ e
4
; E
2
= e
1
− 2e
2
+3e
3
− e
4
; E
3
= e
1
+ e
4
;
E
4
=2e
1
+ e
2
−e
3
+ e
4
.

(DS.





0112
2 −20 1
330−1
1 −11 1





)
13. Cho

21
−12

l`a ma trˆa
.
n chuyˆe
’
nt`u
.
co
.
so

.
’
e
1
,e
2
dˆe
´
nco
.
so
.
’
E
1
, E
2
.T`ım to
.
adˆo
.
cu
’
a c´ac
vecto
.
e
1
, e
2

trong co
.
so
.
’
E
1
, E
2
.
(DS. e
1
=

2
5
,
1
5

. e
2
=

−
1
5
,
2
5


)
Chı
’
dˆa
˜
n. T`u
.
ma trˆa
.
nd˜a cho t`ım khai triˆe
’
n E
1
, E
2
theo co
.
so
.
’
e
1
,e
2
.
T`u
.
d´o t`ım khai triˆe
’

n e
1
,e
2
theo co
.
so
.
’
E
1
, E
2
.
200 Chu
.
o
.
ng 5. Khˆong gian Euclide R
n
14. Cho ma trˆa
.
n



1 −13
51 2
14−1




l`a ma trˆa
.
n chuyˆe
’
nt`u
.
co
.
so
.
’
e
1
,e
2
,e
3
dˆe
´
nco
.
so
.
’
E
1
, E
2

, E
3
.T`ım to
.
adˆo
.
vecto
.
e
2
trong co
.
so
.
’
E
1
, E
2
, E
3
.
(DS. e
2
=

11
41
, −
4

41
, −
5
41

)
15. Cho ma trˆa
.
n



101
002
−131



l`a ma trˆa
.
n chuyˆe
’
nt`u
.
co
.
so
.
’
e

1
,e
2
,e
3
dˆe
´
nco
.
so
.
’
E
1
, E
2
, E
3
.T`ım to
.
adˆo
.
c´ac vecto
.
e
1
,e
2
,e
3

trong co
.
so
.
’
E
1
, E
2
, E
3
.
(D
S. e
1
=

1,
1
3
, 0

, e
2
=

−
1
2
, −

1
3
,
1
2

, e
3
=

0,
1
3
, 0

)
16. Trong co
.
so
.
’
e
1
,e
2
vecto
.
x c´o to
.
ad

ˆo
.
l`a (1; 2). T`ım to
.
adˆo
.
cu
’
a
vecto
.
d
´o trong co
.
so
.
’
E
1
= e
1
+2e
2
; E
2
= −e
1
+ e
2
.

(D
S. x =

−
1
3
, −
4
3

)
17. Trong co
.
so
.
’
e
1
,e
2
vecto
.
x c´o to
.
ad
ˆo
.
l`a ( −3; 1). T`ım to
.
adˆo

.
cu
’
a
vecto
.
d´o trong co
.
so
.
’
E
1
= −2e
1
+ e
2
; E
2
= e
2
.
(D
S. x =

3
2
, −
1
2


)
18. Trong co
.
so
.
’
e
1
,e
2
,e
3
vecto
.
x c´o to
.
ad
ˆo
.
l`a (−1; 2; 0). T`ım to
.
adˆo
.
cu
’
a vecto
.
d´o trong co
.

so
.
’
E
1
=2e
1
− e
2
+3e
3
, E
2
= −3e
1
+ e
2
− 2e
3
;
E
3
=4e
2
+5e
3
.(DS. (−0, 68; −0, 12; 0, 36))
19. Trong co
.
so

.
’
e
1
,e
2
,e
3
vecto
.
x c´o to
.
ad
ˆo
.
l`a (1, −1, 0). T`ım to
.
adˆo
.
cu
’
a vecto
.
d´o trong co
.
so
.
’
: E
1

=3e
1
+ e
2
+6e
3
, E
2
=5e
1
− 3e
2
+7e
3
,
E
3
= −2e
1
+2e
2
− 3e
3
.
5.3. Khˆong gian vecto
.
Euclid. Co
.
so
.

’
tru
.
.
cchuˆa
’
n 201
(DS. x =(−0, 6; 1, 2; 1, 6))
20. Trong co
.
so
.
’
e
1
,e
2
,e
3
vecto
.
x c´o to
.
adˆo
.
l`a (4, 0, −12). T`ım to
.
a
dˆo
.

cu
’
a vecto
.
d
´o trong co
.
so
.
’
E
1
= e
1
+2e
2
+ e
3
, E
2
=2e
1
+3e
2
+4e
3
,
E
3
=3e

1
+4e
2
+3e
3
.
(D
S. x =(−4, −8, 8))
21. Trong khˆong gian v´o
.
imˆo
.
tco
.
so
.
’
l`a e
1
,e
2
,e
3
cho c´ac vecto
.
E
1
=
e
1

+ e
2
, E
2
=2e
1
− e
2
+ e
3
, E
3
= e
2
− e
3
.
1) Ch´u
.
ng minh r˘a
`
ng E
1
, E
2
, E
3
lˆa
.
p th`anh co

.
so
.
’
.
2) T`ım to
.
ad
ˆo
.
cu
’
a vecto
.
x = e
1
+8e
2
− 5e
3
trong co
.
so
.
’
E
1
, E
2
, E

3
.
(D
S. x =(3, −1, 4))
22. Trong co
.
so
.
’
e
1
,e
2
,e
3
cho c´ac vecto
.
a =(1, 2, 3), b =(0, 3, 1),
c =(0, 0, 2), d =(4, 3, 1). Ch´u
.
ng minh r˘a
`
ng c´ac vecto
.
a, b, c lˆa
.
p th`anh
co
.
so

.
’
v`a t`ım to
.
adˆo
.
cu
’
a vecto
.
d trong co
.
so
.
’
d´o.
(D
S. d

4, −
5
3
, −
14
3

)
5.3 Khˆong gian vecto
.
Euclid. Co

.
so
.
’
tru
.
.
c
chuˆa
’
n
Khˆong gian tuyˆe
´
n t´ınh thu
.
.
c V du
.
o
.
.
cgo
.
i l`a khˆong gian Euclid nˆe
´
u trong
V du
.
o
.

.
c trang bi
.
mˆo
.
t t´ıch vˆo hu
.
´o
.
ng, t ´u
.
cl`anˆe
´
uv´o
.
imˆo
˜
ic˘a
.
p phˆa
`
ntu
.
’
x, y ∈Vd
ˆe
`
utu
.
o

.
ng ´u
.
ng v´o
.
imˆo
.
tsˆo
´
thu
.
.
c (k´y hiˆe
.
ul`a x, y) sao cho
∀x, y, z ∈Vv`a sˆo
´
α ∈ R ph´ep tu
.
o
.
ng ´u
.
ng d
´o tho
’
a m˜an c´ac tiˆen dˆe
`
sau
dˆa y

(I) x, y = y,x;
(I I)  x + y,z = x, z + y,z;
(I II) αx, y = αx, y;
(IV) x, x > 0nˆe
´
u x = θ.
Trong khˆong gian vecto
.
R
n
dˆo
´
iv´o
.
ic˘a
.
p vecto
.
a =(a
1
,a
2
, ,a
n
),
202 Chu
.
o
.
ng 5. Khˆong gian Euclide R

n
b =(b
1
,b
2
, ,b
n
) th`ı quy t˘a
´
ctu
.
o
.
ng ´u
.
ng
a, b =
n

i=1
a
i
b
i
= a
1
b
1
+ a
2

b
2
+ ···+ a
n
b
n
(5.12)
s˜e x´ac di
.
nh mˆo
.
t t´ıch vˆo hu
.
´o
.
ng cu
’
a hai vecto
.
a v`a b.
Nhu
.
vˆa
.
y khˆong gian R
n
v´o
.
it´ıchvˆohu
.

´o
.
ng x´ac di
.
nh theo cˆong
th ´u
.
c (5.12) tro
.
’
th`anh khˆong gian Euclid. Do d
´o khi n´oi vˆe
`
khˆong gian
Euclid R
n
ta luˆon luˆon hiˆe
’
u l`a t´ıch vˆo hu
.
´o
.
ng trong d´o x´ac di
.
nh theo
(5.12).
Gia
’
su
.

’
x ∈ R
n
. Khi d´osˆo
´

x, x du
.
o
.
.
cgo
.
il`adˆo
.
d`ai (hay chuˆa
’
n)
cu
’
a vecto
.
x v`a d
u
.
o
.
.
ck´yhiˆe
.

ul`ax.Nhu
.
vˆa
.
y
x
def
=

x, x (5.13)
Vecto
.
x v´o
.
id
ˆo
.
d`ai = 1 du
.
o
.
.
cgo
.
il`ad
u
.
o
.
.

c chuˆa
’
n h´oa hay vecto
.
d
o
.
n
vi
.
.D
ˆe
’
chuˆa
’
n h´oa mˆo
.
t vecto
.
kh´ac θ bˆa
´
tk`y ta chı
’
cˆa
`
n nhˆan n´o v´o
.
isˆo
´
λ =

1
x
.
Dˆo
.
d`ai c´o c´ac t´ınh chˆa
´
t
1
+
x =0⇔ x = θ.
2
+
λx = |λ|x, ∀λ ∈ R.
3
+
|x, y|  xy (bˆa
´
td˘a
’
ng th ´u
.
c Cauchy-Bunhiakovski)
4
+
x + y  x + y (bˆa
´
td˘a
’
ng th´u

.
c tam gi´ac hay bˆa
´
td˘a
’
ng
th ´u
.
c Minkovski).
T`u
.
bˆa
´
td˘a
’
ng th´u
.
c3
+
suy r˘a
`
ng v´o
.
i hai vecto
.
kh´ac θ bˆa
´
tk`yx, y ∈ R
n
ta dˆe

`
uc´o
|x, y|
xcos y
 1 ⇔−1 
x, y
xy
 1.
Sˆo
´
x, y
xy
c´o thˆe
’
xem nhu
.
cosin cu
’
a g´oc ϕ n`ao d´o. G´oc ϕ m`a
cos ϕ =
x, y
xy
, 0  ϕ  π (5.14)
5.3. Khˆong gian vecto
.
Euclid. Co
.
so
.
’

tru
.
.
cchuˆa
’
n 203
du
.
o
.
.
cgo
.
il`ag´oc gi˜u
.
a hai vecto
.
x v`a y.
Hai vecto
.
x, y ∈ R
n
du
.
o
.
.
cgo
.
il`avuˆong g´oc hay tru

.
.
c giao nˆe
´
ut´ıch
vˆo hu
.
´o
.
ng cu
’
ach´ung b˘a
`
ng 0: x, y =0.
Hˆe
.
vecto
.
a
1
,a
2
, ,a
m
∈ R
n
du
.
o
.

.
cgo
.
il`ahˆe
.
tru
.
.
c giao nˆe
´
uch´ung
tru
.
.
c giao t`u
.
ng dˆoi mˆo
.
t, t´u
.
cl`anˆe
´
u a
i
,a
j
 =0∀i = j.
Hˆe
.
vecto

.
a
1
,a
2
, ,a
m
∈ R
n
du
.
o
.
.
cgo
.
il`ahˆe
.
tru
.
.
c giao v`a chuˆa
’
n
h´oa (hay hˆe
.
tru
.
.
c chuˆa

’
n)nˆe
´
u
a
i
,a
i
 = δ
ij
=



0nˆe
´
u i = j
1nˆe
´
u i = j
D
-
i
.
nh l ´y 5.3.1. Mo
.
ihˆe
.
tru
.

.
c giao c´ac vecto
.
kh´ac khˆong d
ˆe
`
ul`ahˆe
.
dˆo
.
c
lˆa
.
p tuyˆe
´
n t´ınh.
Hˆe
.
gˆo
`
m n vecto
.
E
1
, E
2
, ,E
n
∈ R
n

du
.
o
.
.
cgo
.
il`aco
.
so
.
’
tru
.
.
c giao
nˆe
´
u n´o l`a mˆo
.
tco
.
so
.
’
gˆo
`
m c´ac vecto
.
tru

.
.
c giao t`u
.
ng dˆoi mˆo
.
t.
Trong khˆong gian R
n
tˆo
`
nta
.
inh˜u
.
ng co
.
so
.
’
d˘a
.
cbiˆe
.
ttiˆe
.
nlo
.
.
idu

.
o
.
.
c
go
.
i l`a nh˜u
.
ng co
.
so
.
’
tru
.
.
c chuˆa
’
n (vai tr`o nhu
.
co
.
so
.
’
D
ˆec´ac vuˆong g´oc
trong h`ınh ho
.

c gia
’
i t´ıch).
Hˆe
.
gˆo
`
m n vecto
.
E
1
, E
2
, ,E
n
∈ R
n
du
.
o
.
.
cgo
.
i l`a mˆo
.
t co
.
so
.

’
tru
.
.
c
chuˆa
’
n cu
’
a R
n
nˆe
´
u c´ac vecto
.
n`ay t`u
.
ng d
ˆoi mˆo
.
t tru
.
.
c giao v`a d
ˆo
.
d`ai cu
’
a
mˆo

˜
i vecto
.
cu
’
ahˆe
.
dˆe
`
ub˘a
`
ng 1, t´u
.
cl`a
(E
i
, E
k
)=



0nˆe
´
u i = k,
1nˆe
´
u i = k.
D
-

i
.
nh l´y 5.3.2. Trong mo
.
i khˆong gian Euclid n-chiˆe
`
udˆe
`
utˆo
`
nta
.
ico
.
so
.
’
tru
.
.
c chuˆa
’
n.
Dˆe
’
c´o diˆe
`
ud´o ta c´o thˆe
’
su

.
’
du
.
ng ph´ep tru
.
.
c giao h´oa Gram-Smidth
du
.
amˆo
.
tco
.
so
.
’
vˆe
`
co
.
so
.
’
tru
.
.
cchuˆa
’
n. Nˆo

.
i dung cu
’
a thuˆa
.
t to´an d´o n h u
.
sau
Gia
’
su
.
’
E
1
= a
1
.Tiˆe
´
pd´o ph´ep du
.
.
ng du
.
o
.
.
ctiˆe
´
n h`anh theo quy na

.
p.
204 Chu
.
o
.
ng 5. Khˆong gian Euclide R
n
Nˆe
´
u E
1
, E
2
, ,E
i
d˜a d u
.
o
.
.
cdu
.
.
ng th`ı E
i+1
c´o thˆe
’
lˆa
´

y
E
i+1
= a
i+1
+
i

j=1
α
j
a
j
,
trong d´o
α
j
= −
a
i+1
, E
j

E
j
, E
j

,j= 1,i
du

.
o
.
.
ct`ımt`u
.
d
iˆe
`
ukiˆe
.
n E
i+1
tru
.
.
c giao v´o
.
imo
.
i vecto
.
E
1
, E
2
, ,E
i
.
C

´
AC V
´
IDU
.
1. Trong c´ac ph´ep to´an du
.
´o
.
idˆay ph´ep to´an n`ao l`a t´ıch vˆo hu
.
´o
.
ng cu
’
a
hai vecto
.
x =(x
1
,x
2
,x
3
), y =(y
1
,y
2
,y
3

) ∈ R
3
:
1) x, y = x
2
1
y
2
1
+ x
2
2
y
2
2
+ x
2
3
y
2
3
;
2) x, y = x
1
y
1
+2x
2
y
2

+3x
3
y
3
;
3) x, y = x
1
y
1
+ x
2
y
2
−x
3
y
3
.
Gia
’
i. 1) Ph´ep to´an n`ay khˆong l`a t´ıch vˆo hu
.
´o
.
ng v`ı n´o khˆong tho
’
a
m˜an tiˆen d
ˆe
`

III cu
’
a t´ıch vˆo hu
.
´o
.
ng:
αx, y = α
2
x
2
1
y
2
1
+ α
2
x
2
2
y
2
2
+ α
2
x
2
3
y
2

3
= α(x
2
1
y
2
1
+ x
2
2
y
2
2
+ x
2
3
y
2
3
)
2) Ph´ep to´an n`ay l`a t´ıch vˆo hu
.
´o
.
ng. Thˆa
.
tvˆa
.
y, hiˆe
’

n nhiˆen c´ac tiˆen
d
ˆe
`
I v`a I I tho
’
a m˜an. Ta kiˆe
’
m tra c´ac tiˆen dˆe
`
III v`a IV.
Gia
’
su
.
’
x

=(x

1
,x

2
,x

3
), x

=(x


1
,x

2
,x

3
) ∈ R
3
. Khi d´o
x

+ x

,y =(x

1
+ x

1
)y
1
+2(x

2
+ x

2
)y

2
+3(x

3
+ x

3
)y
3
=(x

1
y
1
+2x

2
y
2
+3x

3
y
3
)+(x

1
y
1
+2x


2
y
2
+3x

3
y
3
)
= x

,y+ x

,y.
Tiˆe
´
p theo ta x´et
x, x = x
2
1
+2x
2
2
+3x
2
3
 0v`a
x, x =0⇔ x
2

1
+2x
2
2
+3x
2
3
=0⇔ x
1
= x
2
= x
3
=0⇔ x = θ.
5.3. Khˆong gian vecto
.
Euclid. Co
.
so
.
’
tru
.
.
cchuˆa
’
n 205
V´ı du
.
2. T`ım dˆo

.
d`ai c´ac ca
.
nh v`a g´oc trong ta
.
i A cu
’
a tam gi´ac v´o
.
i
dı
’
nh A(2, 1, −2, −3), B(2, −1, 2, 1) v`a C(6, 5, −2, −1).
Gia
’
i. Ta t`ım to
.
adˆo
.
cu
’
a c´ac vecto
.
−→
AB,
−→
AC v`a
−→
BC.Tac´o
−→

AB(0, −2, 4, 4),
−→
AC(4, 4, 0, 2),
−→
BC(4, 6, −4, −2).
´
Ap du
.
ng d
i
.
nh ngh˜ıa dˆo
.
d`ai vecto
.
trong co
.
so
.
’
tru
.
.
cchuˆa
’
n ta c´o

−→
AB =


0
2
+(−2)
2
+4
2
+4
2
=
√
36 = 6
v`a tu
.
o
.
ng tu
.
.

−→
AC =6,
−→
BC =6
√
2. Theo cˆong th´u
.
c (5.14) ta c´o
cos A =

−→

AB,
−→
AC
AB·AC
=
0 ·4+(−2) ·4+4· 0+4· 2
6 ·6
=0.
Do d
´o

A =
π
2
. 
V´ı d u
.
3. Ch´u
.
ng minh r˘a
`
ng trong bˆa
´
td˘a
’
ng th´u
.
c Cauchy-Bunhiakovski
|a, b|  a·b dˆa
´

ub˘a
`
ng “=” d
a
.
tdu
.
o
.
.
c khi v`a chı
’
khi a v`a b phu
.
thuˆo
.
c tuyˆe
´
n t´ınh.
Gia
’
i. 1) Nˆe
´
u a = λb th`ı
|a, b| = |λb, b = |λ|b
2
= λb·b = ab.
Ngu
.
o

.
.
cla
.
i, nˆe
´
u |a, b| = ab th`ı

a −
a, b
b
2
b, a −
a, b
b
2
b

= a
2
− 2
a, b
b
2
a, b +
a, b
2
b
4
b

2
=
= a
2
− 2
a
2
b
2
b
2
+
a
2
b
2
b
2
b
4
=0.
Nhu
.
ng t´ıch vˆo hu
.
´o
.
ng x, x =0⇔ x = θ.T`u
.
d´o suy ra r˘a

`
ng a =
a, b
b
2
b,t´u
.
cl`aa, b phu
.
thuˆo
.
c tuyˆe
´
n t´ınh. 
V´ı d u
.
4. Hˆe
.
c´ac vecto
.
do
.
nvi
.
trong R
n
v´o
.
i t´ıch vˆo hu
.

´o
.
ng (5.12)
e
1
=(1, 0, 0, ,0)
e
2
=(0, 1, 0, ,0)

e
n
=(0, 0, 0, ,1)
206 Chu
.
o
.
ng 5. Khˆong gian Euclide R
n
l`a mˆo
.
tv´ıdu
.
vˆe
`
co
.
so
.
’

tru
.
.
c chuˆa
’
n trong R
n
.Co
.
so
.
’
n`ay go
.
il`aco
.
so
.
’
ch´ınh t˘a
´
c trong R
n
.
Gia
’
i. Hiˆe
’
n nhiˆen e
i

,e
j
 =0∀i = j, e
j
 =1∀j = 1,n.T`u
.
d
´o
thu d
u
.
o
.
.
cd
iˆe
`
ucˆa
`
nch´u
.
ng minh. 
V´ı d u
.
5. To
.
ad
ˆo
.
cu

’
a vecto
.
a ∈ R
n
bˆa
´
tk`ydˆo
´
iv´o
.
ico
.
so
.
’
tru
.
.
cchuˆa
’
n
b˘a
`
ng t´ıch vˆo hu
.
´o
.
ng cu
’

a vecto
.
d
´o v ´o
.
i vecto
.
co
.
so
.
’
tu
.
o
.
ng ´u
.
ng.
Gia
’
i. Gia
’
su
.
’
a ∈ R
n
v`a E
1

, E
2
, ,E
n
l`a mˆo
.
tco
.
so
.
’
tru
.
.
cchuˆa
’
ncu
’
a
R
n
. Khi d´o
a =
n

i=1
λ
i
E
i

.
Nhˆan vˆo hu
.
´o
.
ng d
˘a
’
ng th ´u
.
c n`ay v´o
.
i E
k
, k =1, 2, ,n ta thu du
.
o
.
.
c
a, E
k
 = λ
k
,k=1, 2, ,n.
Do d´o
a =
n

i=1

a, E
i
E
i
∀a ∈ R
n
.
Sˆo
´
λ
k
= a, E
k
 k =1, 2, ,n ch´ınh l`a to
.
adˆo
.
cu
’
a vecto
.
a ∈ R
n
theo
co
.
so
.
’
tru

.
.
cchuˆa
’
nd˜a cho. 
V´ı d u
.
6. 1) Trong khˆong gian R
3
v´o
.
it´ıchvˆohu
.
´o
.
ng (5.12) cho co
.
so
.
’
E
1
=(1, 2, 1); E
2
=(1, 1, 0); E
3
=(2, 0, 0). H˜ay d`ung phu
.
o
.

ng ph´ap
tru
.
.
c giao h´oa dˆe
’
t`ım co
.
so
.
’
tru
.
.
c giao trong R
3
t`u
.
co
.
so
.
’
d˜a cho.
2) Trong khˆong gian R
3
v´o
.
i t´ıch vˆo hu
.

´o
.
ng (5.12) cho co
.
so
.
’
E
1
=
(1, −1, 1), E
2
=(2, −3, 4), E
3
=(2, 2, 6). H˜ay du
.
.
ng co
.
so
.
’
tru
.
.
cchuˆa
’
n
trong R
3

theo co
.
so
.
’
d˜a cho.
Gia
’
i. 1) Tru
.
´o
.
chˆe
´
t ta cho
.
n E
1
= E
1
=(1, 2, 1). Tiˆe
´
p theo d˘a
.
t
E
2
= E
2
+ λE

1
sao cho E
2
,E
1
 =0,t´u
.
cl`a
E
2
,E
1
 = E
1
, E
2
 + λE
1
,E
1
 =0.
5.3. Khˆong gian vecto
.
Euclid. Co
.
so
.
’
tru
.

.
cchuˆa
’
n 207
Nhu
.
ng E
1
,E
1
= 0 (cu
.
thˆe
’
l`a > 0) v`ı E
1
= E
1
= θ.Dod´o
λ = −
E
1
, E
2

E
1
,E
1


= −
(1, 2, 1), (1, 1, 0)
1
2
+2
2
+1
2
= −
1
2
·
Do d
´o
E
2
=(1, 1, 0) −
1
2
(1, 2, 1) =

1
2
, 0, −
1
2

.
Tiˆe
´

p theo d
˘a
.
t
E
3
= E
3
+ αE
1
+ βE
2
sao cho E
3
,E
1
 = E
3
,E
2
 =0. Tu
.
o
.
ng tu
.
.
nhu
.
trˆen, t`u

.
diˆe
`
ukiˆe
.
n
E
3
,E
1
 =0tac´oα = −
1
3
v`a t`u
.
d
iˆe
`
ukiˆe
.
n E
3
,E
2
 =0tac´oβ = −2.
Do d´o
E
3
= E
3

−
1
3
E
1
− 2E
2
=

2
3
, −
2
3
,
2
3

v`a thu d
u
.
o
.
.
cco
.
so
.
’
tru

.
.
c giao
E
1
=(1, 2, 1),E
2
=

1
2
, 0, −
1
2

,E
3
=

2
3
, −
2
3
,
2
3

.
2) Tu

.
o
.
ng tu
.
.
nhu
.
phˆa
`
n 1), dˆa
`
u tiˆen ta d˘a
.
t
E
1
= E
1
=(1, −1, 1)
E
2
= E
2
+ λE
1
sao cho E
2
,E
1

 =0.T`u
.
d´othudu
.
o
.
.
c
λ = −
E
1
, E
2

E
1
,E
1

= −
2+3+4
3
= −3,
v`a do d
´o
E
2
=(−1, 0, 1).
208 Chu
.

o
.
ng 5. Khˆong gian Euclide R
n
Tiˆe
´
p theo ta t`ım
E
3
= E
3
+ αE
1
+ βE
2
sao cho E
3
,E
1
 =0,E
3
,E
2
 =0v`at`u
.
d
´o t h u d u
.
o
.

.
c
α = −
E
1
, E
3

E
1
,E
1

= −2; β = −
E
2
, E
3

E
2
,E
2

= −2.
Nhu
.
vˆa
.
y

E
3
=(2, 4, 2).
Sau c`ung ta chuˆa
’
n h´oa c´ac vecto
.
E
1
,E
2
,E
3
v`a thu du
.
o
.
.
cco
.
so
.
’
tru
.
.
c
chuˆa
’
n

e
1
=

1
√
3
, −
1
√
3
,
1
√
3

,e
2
=

−
1
√
2
, 0,
1
√
2

,

e
3
=

1
√
6
,
2
√
6
,
1
√
6

. 
V´ı du
.
7. H˜ay bˆo
’
sung cho hˆe
.
tru
.
.
c giao gˆo
`
m ba vecto
.

trong R
4
:
b
1
=(1, 1, 1, 1),b
2
=(2, 2, −2, −2),b
3
=

−
1
2
,
1
2
, −
7
2
,
7
2

d
ˆe
’
thu du
.
o

.
.
cco
.
so
.
’
tru
.
.
c giao trong khˆong gian d
´o.
Gia
’
i. Ta c´o thˆe
’
bˆo
’
sung b˘a
`
ng hai c´ach
1
+
V`ısˆo
´
vecto
.
cu
’
ahˆe

.
d˜a cho nho
’
ho
.
n 4 (l`a sˆo
´
chiˆe
`
ucu
’
a khˆong
gian R
4
)nˆen trong khˆong gian R
4
ta c´o thˆe
’
cho
.
n vecto
.
a
4
sao cho hˆe
.
vecto
.
b
1

,b
2
,b
3
,a
4
dˆo
.
clˆa
.
p tuyˆe
´
n t´ınh v`a sau d´o ´ap du
.
ng ph´ep tru
.
.
c giao
h´oa Gram-Smidth.
2
+
Tac´othˆe
’
cho
.
n vecto
.
x =(x
1
,x

2
,x
3
,x
4
)dˆo
`
ng th`o
.
i tru
.
.
c giao v´o
.
i
c´ac vecto
.
b
1
,b
2
,b
3
,t´u
.
c l`a thu du
.
o
.
.

chˆe
.
phu
.
o
.
ng tr`ınh
x
1
+ x
2
+ x
3
+ x
4
=0,
2x
1
+2x
2
− 2x
3
−2x
4
=0,
−
1
2
x
1

+
1
2
x
2
−
7
2
x
3
+
7
2
x
4
=0.
5.3. Khˆong gian vecto
.
Euclid. Co
.
so
.
’
tru
.
.
cchuˆa
’
n 209
Ch˘a

’
ng ha
.
n, t`u
.
hˆe
.
d´o ta c´o x =(7, −7, −1, 1). 
V´ı du
.
8. 1
+
Ch´u
.
ng to
’
r˘a
`
ng c´ac vecto
.
x
1
=(1, 1, 1, 2) v`a x
2
=
(1, 2, 3, −3) l`a tru
.
.
c giao v´o
.

i nhau.
2
+
H˜ay bˆo
’
sung cho hˆe
.
hai vecto
.
d´o d ˆe
’
thu du
.
o
.
.
cco
.
so
.
’
tru
.
.
c giao
cu
’
a R
4
.

Gia
’
i. 1
+
Ta c´o
x
1
,x
2
 =1· 1+1· 2+1· 3 −2 · 3=0.
Do d
´och´ung tru
.
.
c giao.
2
+
Gia
’
su
.
’
x
3
=(α,β,γ,0), trong d´o α,β,γ du
.
o
.
.
cx´acd

i
.
nh t `u
.
c´ac
d
iˆe
`
ukiˆe
.
n x
3
,x
1
 =0,x
3
,x
2
 =0t´u
.
cl`a
α + β + γ =0
α +2β +3γ =0.
T`u
.
d´o x
3
=(1, −2, 1, 0).
Bˆay gi`o
.

ta s˜e bˆo
’
sung thˆem cho hˆe
.
vecto
.
x
1
,x
2
,x
3
mˆo
.
t vecto
.
n˜u
.
a.
Gia
’
su
.
’
x
4
=(α,β,γ,δ), trong d´o c´ac to
.
adˆo
.

α, β, γ, δ du
.
o
.
.
c x´ac d
i
.
nh
t`u
.
c´ac d
˘a
’
ng th ´u
.
c:
x
4
,x
1
 =0, x
4
,x
2
 =0, x
4
,x
2
 =0.

T`u
.
d´o
α + β + γ +2δ =0,
α + β +3γ − 3δ =0,
α −2β + γ =0.
T`u
.
d´othudu
.
o
.
.
c x
4
=(−25, −4, 17, 6). Nhu
.
vˆa
.
ytad˜abˆo
’
sung thˆem
hai vecto
.
x
3
,x
4
v`a thu du
.

o
.
.
chˆe
.
vecto
.
tru
.
.
c giao x
1
,x
2
,x
3
,x
4
trong
khˆong gian 4-chiˆe
`
u. D´o l `a c o
.
so
.
’
tru
.
.
c giao. 

B
`
AI T
ˆ
A
.
P
210 Chu
.
o
.
ng 5. Khˆong gian Euclide R
n
1. Gia
’
su
.
’
a =(a
1
,a
2
), b =(b
1
,b
2
)l`anh˜u
.
ng vecto
.

t`uy ´y cu
’
a R
2
. Trong
c´ac quy t˘a
´
c sau dˆay, quy t˘a
´
c n`ao x´ac di
.
nh t´ıch vˆo hu
.
´o
.
ng trˆen R
2
:
1) a, b = a
1
b
1
+ a
2
b
2
.
2) a, b = ka
1
b

1
+ a
2
b
2
, k, =0.
3) a, b = a
1
b
1
+ a
1
b
2
+ a
2
b
1
.
4) a, b =2a
1
b
1
+ a
1
b
2
+ a
2
b

1
+ a
2
b
2
.
5) a, b =3a
1
b
1
+ a
1
b
2
+ a
2
b
1
−a
2
b
2
.
(D
S. 1), 2) v`a 4) x´ac di
.
nh t´ıch vˆo hu
.
´o
.

ng
3) v`a 5) khˆong x´ac di
.
nh t´ıch vˆo hu
.
´o
.
ng).
2. Trong khˆong gian Euclide R
4
,x´acdi
.
nh g´oc gi˜u
.
a c´ac vecto
.
:
1) a =(1, 1, 1, 1), b =(3, 5, 1, 1). (D
S. arccos
5
6
)
2) a =(1, 1, 1, 1), b =(3, −5, 1, 1). (D
S.
π
2
)
3) a =(1, 1, 1, 1), b =(−3, −3, −3, −3). (D
S. π)
3. Trong khˆong gian Euclid R

4
,t`ımdˆo
.
d`ai cu
’
a c´ac ca
.
nh v`a c´ac g´oc
cu
’
a tam gi´ac lˆa
.
pbo
.
’
i c´ac vecto
.
a, b, a + b nˆe
´
u
1) a v`a b nhu
.
trong 2.1)
2) a v`a b nhu
.
trong 2.2)
3) a =(2, −1, 2, 4), b =(2, −1, 2, −4).
(DS. 1) a =2,b =6,a + b =2
√
15, cos(


a, b)=
5
6
,
cos(

a, a + b)=
7
2
√
15
; cos(

b, a + b)=
13
6
√
15
;2)a =2,b =6,
a + b =2
√
10, cos(

a, b) = 0, cos(

a, a + b)=
1
√
10

, cos(

b, a + b)=
3
√
10
;3)a =5,b =5,a + b = 6, cos(

a, b)=−
7
25
,
cos(

a, a + b)=
4
5
, cos(

b, a + b)=
4
15
)
4. Ch´u
.
ng minh r˘a
`
ng trong khˆong gian Euclide
1) a ⊥ a ⇔ a = θ.
2) Nˆe

´
u vecto
.
a ⊥ b
i
∀i = 1,s th`ı a tru
.
.
c giao v´o
.
imo
.
itˆo
’
ho
.
.
p tuyˆe
´
n
t´ınh cu
’
a b
1
, ,b
s
.
5.3. Khˆong gian vecto
.
Euclid. Co

.
so
.
’
tru
.
.
cchuˆa
’
n 211
3) Hˆe
.
c´ac vecto
.
kh´ac khˆong v`a tru
.
.
c giao v´o
.
i nhau t`u
.
ng dˆoi mˆo
.
tl`a
hˆe
.
dˆo
.
clˆa
.

p tuyˆe
´
n t´ınh.
5. Gia
’
su
.
’
mˆo
.
t tam gi´ac trong khˆong gian Euclide d
u
.
o
.
.
clˆa
.
pnˆenbo
.
’
i
c´ac vecto
.
a, b, a + b.Ch´u
.
ng minh:
1) di
.
nh l´y Pithago: Nˆe

´
u a ⊥ b ⇒a + b
2
= a
2
+ b
2
.
2) di
.
nh l´y da
’
ocu
’
adi
.
nh l´y Pithago: Nˆe
´
u a + b
2
= a
2
+ b
2
⇒
a ⊥ b.
3) d
i
.
nh l´y h`am cosin:

a + b
2
= a
2
+ b
2
+2abcos(

a, b).
4) bˆa
´
td˘a
’
ng th ´u
.
c tam gi´ac
a−b  a + b≤a+ b.
Chı
’
dˆa
˜
n. Su
.
’
du
.
ng bˆa
´
td˘a
’

ng th´u
.
c Cauchy-Bunhiakovski.
6. Ch´u
.
ng minh r˘a
`
ng trong h`ınh b`ınh h`anh du
.
.
ng trˆen hai vecto
.
a v`a b
tˆo
’
ng c´ac b`ınh phu
.
o
.
ng dˆo
.
d`ai cu
’
a c´ac du
.
`o
.
ng ch´eo b˘a
`
ng tˆo

’
ng c´ac b`ınh
phu
.
o
.
ng d
ˆo
.
d`ai c´ac ca
.
nh
a + b
2
+ a − b
2
=2a
2
+2b
2
.
7. Ch´u
.
ng minh r˘a
`
ng nˆe
´
u c´ac vecto
.
a

1
,a
2
, ,a
m
cu
’
a khˆong gian
Euclide l`a t`u
.
ng dˆoi mˆo
.
t tru
.
.
c giao th`ı
a
1
+ a
2
+ ···+ a
m

2
= a
1

2
+ a
2


2
+ ···+ a
m

2
.
Chı
’
dˆa
˜
n. X´et t´ıch vˆo hu
.
´o
.
ng
a
1
+ a
2
+ ···+ a
m
,a
1
+ a
2
+ ···+ a
m

8.

´
Ap du
.
ng qu´a tr`ınh tru
.
.
c giao h´oa d
ˆo
´
iv´o
.
i c´ac hˆe
.
vecto
.
sau d
ˆay cu
’
a
R
n
:
1) a
1
=(1, −2, 2), a
2
=(−1, 0, −1), a
3
=(5, −3, −7).
212 Chu

.
o
.
ng 5. Khˆong gian Euclide R
n
(DS. E
1
= a
1
=(1, −2, 2); E
2
=

−
2
3
, −
2
3
, −
1
3

; E
3
=(6, −3, −6))
2) a
1
=(1, 1, 1, 1), a
2

=(3, 3, −1, −1), a
3
=(−2, 0, 6, 8).
(D
S. E
1
= a
1
=(1, 1, 1, 1); E
2
=(2, 2, −2, −2), E
3
=(−1, 1, −1, 1))
3) a
1
=(1, 1, 1, 1); a
2
=(3, 3, −1, −1); a
3
=(−1, 0, 3, 4).
(DS. E
1
= a
1
=(1, 1, 1, 1), E
2
=(2, 2, −2, −2), E
3
=


−
1
2
,
1
2
, −
7
2
,
7
2

)
9. Tru
.
.
cchuˆa
’
n h´oa c´ac hˆe
.
vecto
.
sau dˆay cu
’
a khˆong gian R
4
:
1) a
1

=(1, 1, 1, 1), a
2
=(1, 1, −3, −3), a
3
=(4, 3, 0, −1).
(D
S. E
1
=

1
2
,
1
2
,
1
2
,
1
2

, E
2
=

1
2
,
1

2
, −
1
2
, −
1
2

, E
3
=

1
2
, −
1
2
,
1
2
, −
1
2

)
2) a
1
=(1, 2, 2, 0), a
2
=(1, 1, 3, 5), a

3
=(1, 0, 1, 0).
(D
S. E
1
=

1
3
,
2
3
,
2
3
, 0

, E
2
=

0, −
1
3
√
3
,
1
3
√

3
,
5
3
√
3

,
E
3
=

6
√
78
, −
17
3
√
78
,
8
3
√
78
, −
5
3
√
78


)
10. Ch´u
.
ng to
’
r˘a
`
ng c´ac hˆe
.
vecto
.
sau dˆay trong R
4
l`a tru
.
.
c giao v`a bˆo
’
sung cho c´ac hˆe
.
d
´odˆe
’
tro
.
’
th`anh co
.
so

.
’
tru
.
.
c giao:
1) a
1
=(1, −2, 1, 3), a
2
=(2, 1, −3, 1)
(DS. Ch˘a
’
ng ha
.
n, c´ac vecto
.
a
3
=(1, 1, 1, 0), a
4
=(−1, 1, 0, 1))
2) a
1
=(1, −1, 1, −3), a
2
=(−4, 1, 5, 0).
(D
S. Ch˘a
’

ng ha
.
n, c´ac vecto
.
a
3
=(2, 3, 1, 0) v`a a
4
=(1, −1, 1, 1))
11. Ch´u
.
ng to
’
r˘a
`
ng c´ac vecto
.
sau d
ˆay trong R
4
l`a tru
.
.
c giao v`a bˆo
’
sung
cho c´ac hˆe
.
d
´o d ˆe

’
tro
.
’
th`anh co
.
so
.
’
tru
.
.
c giao v`a chuˆa
’
n h´oa c´ac co
.
so
.
’
d
´o
1) a
1
=(1, −1, 1, −1), a
2
=(1, 1, 1, 1).
(D
S. E
1
=


1
2
, −
1
2
,
1
2
, −
1
2

, E
2
=

1
2
,
1
2
,
1
2
,
1
2

, E

3
=

−
1
√
2
, 0,
1
√
2
, 0

,
5.4. Ph´ep biˆe
´
ndˆo
’
i tuyˆe
´
n t´ınh 213
E
4
=

0, −
1
√
2
, 0,

1
√
2

)
2) a
1
=(1, −1, −1, 3), a
2
=(1, 1, −3, −1)
(DS. E
1
=

1
2
√
3
, −
1
2
√
3
, −
1
2
√
3
,
√

3
2

, E
2
=

1
2
√
3
,
1
2
√
3
, −
√
3
2
, −
1
2
√
3

,
E
3
=


2
√
6
,
1
√
6
,
1
√
6
, 0

, E
4
=

−
1
√
6
,
2
√
6
, 0,
1
√
6


)
5.4 Ph´ep biˆe
´
ndˆo
’
i tuyˆe
´
nt´ınh
5.4.1 D
-
i
.
nh ngh˜ıa
´
Anh xa
.
L : R
n
→ R
n
biˆe
´
n khˆong gian R
n
th`anh ch´ınh n´o du
.
o
.
.

cgo
.
il`a
mˆo
.
t ph´ep biˆe
´
nd
ˆo
’
i tuyˆe
´
nt´ınh (bdtt) cu
’
a khˆong gian R
n
nˆe
´
u n´o tho
’
a
m˜an hai diˆe
`
ukiˆe
.
n sau dˆa y
(i) V´o
.
i hai vecto
.

a v`a b ∈ R
n
bˆa
´
tk`y
L(a + b)=L(a)+L(b). (5.15)
(ii) V´o
.
i vecto
.
a ∈ R
n
bˆa
´
tk`yv`a∀λ ∈ R ta c´o
L(λa)=λL(a). (5.16)
Hai d
iˆe
`
ukiˆe
.
n (5.15) v`a (5.16) tu
.
o
.
ng du
.
o
.
ng v´o

.
idiˆe
`
ukiˆe
.
n:
L(λ
1
a + λ
2
b)=λ
1
L(a)+λ
2
L(b).
T`u
.
di
.
nh ngh˜ıa suy ra: nˆe
´
uhˆe
.
vecto
.
a
1
,a
2
, ,a

m
∈ R
n
l`a pttt th`ı
hˆe
.
c´ac vecto
.
a
’
nh f(a
1
), ,f(a
m
)c˜ung l`a pttt.
5.4.2 Ma trˆa
.
ncu
’
a ph´ep bdtt
Gia
’
su
.
’
trong khˆong gian R
n
ta cˆo
´
di

.
nh mˆo
.
tco
.
so
.
’
(E) n`ao d´o:
E = {E
1
, E
2
, ,E
n
}. (5.17)
214 Chu
.
o
.
ng 5. Khˆong gian Euclide R
n
Khi d´o ∀x ∈ R
n
ta c´o khai triˆe
’
n
x = x
1
E

1
+ x
2

2
+ ···+ x
n
E
n
.
Mo
.
i ma trˆa
.
n vuˆong A =


a
ij


n×n
dˆe
`
u x´ac di
.
nh ph´ep bdtt L theo
cˆong th´u
.
c






y
1
y
2
···
y
n





=






a
11
a
12
a
1n

a
21
a
22
a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
n1
a
n2
a
nn













x
1
x
2
.
.
.
x
n






(5.18)
N´oi c´ach kh´ac: d
ˆe
’
thu du
.
o
.
.

cto
.
ad
ˆo
.
a
’
nh y = L(x) ta cˆa
`
n nhˆan ma trˆa
.
n
A v´o
.
icˆo
.
tto
.
adˆo
.
cu
’
a x.Viˆe
´
t ra chi tiˆe
´
t ta c´o
y
1
= a

11
x
1
+ a
12
x
2
+ ···+ a
1n
x
n
,
(5.19)
y
n
= a
n1
x
1
+ a
n2
x
2
+ ···+ a
nn
x
n
.
Ngu
.

o
.
.
cla
.
i, trong co
.
so
.
’
d
˜a c h o
.
n (5.17) mˆo
˜
i ph´ep bdtt L dˆe
`
utu
.
o
.
ng
´u
.
ng v´o
.
imˆo
.
t ma trˆa
.

n A = a
ij
 cˆa
´
p n v`a su
.
.
t´ac dˆo
.
ng cu
’
a ph´ep bdtt
du
.
o
.
.
c thu
.
.
chiˆe
.
n theo cˆong th´u
.
c (5.18) hay (5.19).
Viˆe
.
c t`ım ma trˆa
.
ncu

’
aph´epbdtt du
.
o
.
.
ctiˆe
´
n h`anh nhu
.
sau
1
+
T´ac dˆo
.
ng L lˆen c´ac vecto
.
co
.
so
.
’
cu
’
a (5.17) v`a thu du
.
o
.
.
ca

’
nh
L(E
i
), i = 1,n.
2
+
Khai triˆe
’
n c´ac a
’
nh L(E
i
) theo co
.
so
.
’
(5.17):
L(E
1
)=a
11
E
1
+ a
21
E
2
+ ···+ a

n1
E
n
,
L(E
2
)=a
12
E
1
+ a
22
E
2
+ ···+ a
n2
E
n
,

L(E
\
)=a
1n
E
1
+ a
2n
E
2

+ ···+ a
nn
E
n
.









(5.20)
T`u
.
c´ac to
.
adˆo
.
trong (5.20) ta lˆa
.
p ma trˆa
.
n A sao cho to
.
adˆo
.
cu

’
a vecto
.
5.4. Ph´ep biˆe
´
ndˆo
’
i tuyˆe
´
n t´ınh 215
L(E
i
), i = 1,n l`a cˆo
.
tth´u
.
i cu
’
a A,t´u
.
cl`a
A =






a
11

a
12
a
1n
a
21
a
22
a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a
n1
a
n2
a
nn







D
´o l`a ma trˆa
.
ncu
’
aph´epbdtt.
Ta lu
.
u´yr˘a
`
ng khi thay d
ˆo
’
ico
.
so
.
’
th`ı ma trˆa
.
ncu
’
a ph´ep biˆe
´
nd
ˆo

’
i
tuyˆe
´
n t´ınh s˜e thay dˆo
’
i. Gia
’
su
.
’
ma trˆa
.
n chuyˆe
’
nt`u
.
co
.
so
.
’
(E)dˆe
´
nco
.
so
.
’
(E


)du
.
o
.
.
ck´yhiˆe
.
ul`aT
EE

, trong d´o
E

= {E

1
, E

2
, ,E

n
}
v`a A l`a ma trˆa
.
n ph´ep biˆe
´
ndˆo
’

i tuyˆe
´
n t´ınh L theo co
.
so
.
’
(5.17). Khi d
´o,
ma trˆa
.
n B cu
’
a L theo co
.
so
.
’
(E

)liˆen hˆe
.
v´o
.
i ma trˆa
.
n A cu
’
a n´o theo
co

.
so
.
’
(5.17) bo
.
’
i cˆong th´u
.
c
B = T
−1
EE

AT
EE

(5.21)
hay l`a
A = T
EE

BT
−1
EE

(5.22)
5.4.3 C´ac ph´ep to´an
Gia
’

su
.
’
A v`a B l`a hai ph´ep bdtt cu
’
a khˆong gian R
n
v´o
.
i ma trˆa
.
ntu
.
o
.
ng
´u
.
ng l`a A = a
ij
 v`a B = b
ij
 t`uy ´y.
1
+
Tˆo
’
ng A + B l`a ph´ep biˆe
´
ndˆo

’
i C sao cho
C(x)=A(x)+B(x) ∀x ∈ R
n
v´o
.
i ma trˆa
.
ntu
.
o
.
ng ´u
.
ng l`a C = A + B = a
ij
 + b
ij
.
216 Chu
.
o
.
ng 5. Khˆong gian Euclide R
n
2
+
T´ıch c´ac ph´ep biˆe
´
ndˆo

’
i tuyˆe
´
n t´ınh A v´o
.
isˆo
´
thu
.
.
c α ∈ R l`a ph´ep
biˆe
´
ndˆo
’
i αA sao cho
(αA)(x)=αA(x)
v´o
.
i ma trˆa
.
nl`aαa
ij
.
3
+
T´ıch AB l`a ph´ep biˆe
´
ndˆo
’

i
C(x)=A(B(x))
v´o
.
i ma trˆa
.
nl`aC = A · B.
5.4.4 Vecto
.
riˆeng v`a gi´a tri
.
riˆeng
Vecto
.
kh´ac khˆong x ∈ R
n
du
.
o
.
.
cgo
.
il`avecto
.
riˆeng cu
’
a ph´ep biˆe
´
ndˆo

’
i
tuyˆe
´
n t´ınh L nˆe
´
ut`ımdu
.
o
.
.
csˆo
´
λ sao cho d˘a
’
ng th´u
.
c sau tho
’
a m˜an
L(x)=λx. (5.23)
Sˆo
´
λ d
u
.
o
.
.
cgo

.
il`agi´a tri
.
riˆeng cu
’
aph´epbdtt L tu
.
o
.
ng ´u
.
ng v´o
.
i vecto
.
riˆeng x.
C´ac t´ınh chˆa
´
tcu
’
a vecto
.
riˆeng
1
+
Mˆo
˜
i vecto
.
riˆeng chı

’
c´o mˆo
.
t gi´a tri
.
riˆeng.
2
+
Nˆe
´
u x
1
v`a x
2
l`a c´ac vecto
.
riˆeng cu
’
aph´epbdtt L v´o
.
ic`ung mˆo
.
t
gi´a tri
.
riˆeng λ th`ı tˆo
’
ng x
1
+ x

2
c˜ung l`a vecto
.
riˆeng cu
’
a L v´o
.
i gi´a tri
.
riˆeng λ.
3
+
Nˆe
´
u x l`a vecto
.
riˆeng cu
’
a L v´o
.
i gi´a tri
.
riˆeng λ th`ı mo
.
i vecto
.
αx
(α =0)c˜ung l`a vecto
.
riˆeng cu

’
a L v´o
.
i gi´a tri
.
riˆeng λ.
Nˆe
´
u trong khˆong gian R
n
d˜acho
.
nmˆo
.
tco
.
so
.
’
x´ac d
i
.
nh th`ı (5.23) c´o
thˆe
’
viˆe
´
tdu
.
´o

.
ida
.
ng ma trˆa
.
n
AX = λX (5.24)
v`a khi d
´o: mo
.
icˆo
.
t kh´ac khˆong tho
’
a m˜an (5.24) du
.
o
.
.
cgo
.
il`avecto
.
riˆeng
cu
’
a ma trˆa
.
n A tu
.

o
.
ng ´u
.
ng v´o
.
i gi´a tri
.
riˆeng λ.
5.4. Ph´ep biˆe
´
ndˆo
’
i tuyˆe
´
n t´ınh 217
V`ı λX = λEX, trong d´o E l`a ma trˆa
.
ndo
.
nvi
.
nˆen (5.24) c´o thˆe
’
viˆe
´
tdu
.
´o
.

ida
.
ng
(A −λE)X =0
v`a du
.
´o
.
ida
.
ng to
.
ad
ˆo
.
ta thu du
.
o
.
.
c
(a
11
− λ)x
1
+ a
12
x
2
+ ···+ a

1n
x
n
=0,
a
21
x
1
+(a
22
− λ)x
2
+ ···+ a
2n
x
n
=0,

a
n1
x
1
+ a
n2
x
2
+ ···+(a
nn
−λ)x
n

=0.









(5.25)
D
ˆe
’
t`ım c´ac vecto
.
riˆeng, tru
.
´o
.
chˆe
´
tcˆa
`
n t`ım nghiˆe
.
m kh´ac 0 cu
’
ahˆe
.

(5.25). Nghiˆe
.
m kh´ac 0 cu
’
ahˆe
.
(5.25) tˆo
`
nta
.
i khi v`a chı
’
khi di
.
nh th ´u
.
c
cu
’
a n´o b˘a
`
ng 0, t ´u
.
cl`a
|A − λE| =











a
11
−λa
12
a
1n
a
21
a
22
− λ a
2n
.
.
.
.
.
.
.
.
.
.
.
.
a

n1
a
n2
a
nn
− λ










=0. (5.26)
Phu
.
o
.
ng tr`ınh (5.26) du
.
o
.
.
cgo
.
il`aphu
.

o
.
ng tr`ınh d˘a
.
c tru
.
ng cu
’
ama
trˆa
.
n A, c`on c´ac nghiˆe
.
mcu
’
a n´o go
.
i l`a c´ac sˆo
´
d˘a
.
c tru
.
ng hay gi´a tri
.
riˆeng
cu
’
a ma trˆa
.

n A. Sau khi t`ım du
.
o
.
.
c c´ac sˆo
´
d˘a
.
c tru
.
ng λ
1
,λ
2
, ,λ
n
ta
cˆa
`
n thay gi´a tri
.
λ
i
v`ao (5.25) dˆe
’
t`ım c´ac to
.
adˆo
.

x
1
, ,x
n
cu
’
a vecto
.
riˆeng tu
.
o
.
ng ´u
.
ng.
C
´
AC V
´
IDU
.
V´ı d u
.
1. Cho L : R
2
→ R
2
(a
1
,a

2
) −→ L(a
1
,a
2
)=(a
1
+ a
2
, 2a
1
).
1
+
Ch´u
.
ng minh r˘a
`
ng L l`a ph´ep biˆe
´
ndˆo
’
i tuyˆe
´
n t´ınh.
2
+
T`ım ma trˆa
.
ncu

’
a L theo co
.
so
.
’
ch´ınh t˘a
´
c e = {e
1
,e
2
}.
218 Chu
.
o
.
ng 5. Khˆong gian Euclide R
n
Gia
’
i. 1
+
Gia
’
su
.
’
x =(x
1

,x
2
), y =(y
1
,y
2
). Khi d´o
αx + βy = α(x
1
,x
2
)+β(y
1
,y
2
)=(αx
1
+ βy
1
,αx
2
+ βy
2
)
v`a do d
´o
L(αx + βy)=L(αx
1
+ βy
1

,αx
2
+ βy
2
)
=

αx
1
+ βy
1
+ αx
2
+ βy
2
, 2(αx
1
+ βy
1
)

=

α(x
1
+ x
2
)+β(y
1
+ y

2
),α2x
1
+ β2y
1

=

α(x
1
+ x
2
),α2x
1

+

β(y
1
+ y
2
),β2y
1

= α(x
1
+ x
2
, 2x
1

)+β(y
1
+ y
2
, 2y
1
)
= αL(x
1
,x
2
)+βL(y
1
,y
2
)
= αL(x)+βL(y).
Nhu
.
vˆa
.
y L l`a ph´ep bdtt.
2
+
Dˆe
’
t`ım ma trˆa
.
ncu
’

a ph´ep biˆe
´
ndˆo
’
i tuyˆe
´
n t´ınh L ta khai triˆe
’
n
a
’
nh L(e
1
)v`aL(e
2
) theo co
.
so
.
’
ch´ınh t˘a
´
c. Ta c´o
L(e
1
)=L(1, 0) = (1, 2 ·1) = L(1, 2) = 1 · e
1
+2· e
2
,

L(e
2
)=L(0, 1) = (1, 2 ·0) = L(1, 0) = 1 · e
1
+0· e
2
.
T`u
.
d
´othudu
.
o
.
.
c
A =

11
20

. 
V´ı d u
.
2. X´et khˆong gian R
3
v´o
.
ico
.

so
.
’
E: E
1
=(1, 1, 1), E
2
=(0, 1, 1),
E
3
=(0, 0, 1) v`a ph´ep biˆe
´
ndˆo
’
i L : R
3
→ R
3
x´ac di
.
nh bo
.
’
id˘a
’
ng th´u
.
c
L[(u
1

,u
2
,u
3
)] = (u
1
,u
2
− u
1
,u
3
− u
1
) ∀u =(u
1
,u
2
,u
3
) ∈ R
3
.
1
+
Ch´u
.
ng minh r˘a
`
ng L l`a ph´ep bdtt.

2
+
T`ım ma trˆa
.
ncu
’
a L trong co
.
so
.
’
d˜acho
.
n.
5.4. Ph´ep biˆe
´
ndˆo
’
i tuyˆe
´
n t´ınh 219
Gia
’
i. 1
+
Gia
’
su
.
’

x =(x
1
,x
2
,x
3
), y =(y
1
,y
2
,y
3
) ∈ R
3
v`a α, β ∈ R.
Ta c´o
L(αx + βy)=L

α(x
1
,x
2
,x
3
)+β(y
1
,y
2
,y
3

)

= L

(αx
1
+ βy
1
,αx
2
+ βy
2
,αx
3
+ βy
3
)

=

αx
1
+ βy
1
,αx
2
+ βy
2
−αx
1

− βy
1
,αx
3
+ βy
3
− αx
1
− βy
1

=

αx
1
,α(x
2
− x
1
),α(x
3
− x
1
)

+

βy
1
,β(y

2
− y
1
),β(y
3
− y
2
)

= α(x
1
,x
2
−x
1
,x
3
− x
1
)+β(y
1
,y
2
− y
1
,y
3
− y
1
)

= αL(x)+βL(y).
Vˆa
.
y L l`a ph´ep biˆe
´
nd
ˆo
’
i phˆan tuyˆe
´
n t´ınh.
2
+
Dˆe
’
t`ım ma trˆa
.
ncu
’
a L dˆo
´
iv´o
.
ico
.
so
.
’
E
1

, E
2
, E
3
ta c´o
L(E
1
)=L(1, 1, 1) = (1, 1 − 1, 1 − 1) = (1, 0, 0) = E
1
+0·E
2
+0·E
3
,
L(E
2
)=L(0, 1, 1) = (0, 1, 1) = 0 ·E
1
+1·E
2
+1·E
3
,
L(E
3
)=L(0, 0, 1) = (0, 0, 1) = 0 ·E
1
+0·E
2
+1·E

3
.
T`u
.
d
´o suy r˘a
`
ng ma trˆa
.
ncu
’
a L dˆo
´
iv´o
.
ico
.
so
.
’
d
˜achol`a
A =



100
010
011




. 
V´ı du
.
3. Trong khˆong gian R
3
cho co
.
so
.
’
ch´ınh t˘a
´
c e = {e
1
,e
2
,e
3
} v`a
E = {E
1
, E
2
, E
3
}, E
1
=2e

1
− e
2
+3e
3
, E
2
= e
1
+ e
3
, E
3
= −e
2
+2e
3
l`a
mˆo
.
tco
.
so
.
’
kh´ac v`a gia
’
su
.
’

L : R
3
→ R
3
l`a ´anh xa
.
du
.
o
.
.
cx´acdi
.
nh theo
co
.
so
.
’
{e
1
,e
2
,e
3
} nhu
.
sau
x =(x
1

,x
2
,x
3
) −→ f(x)=(x, x
1
+ x
2
,x
1
+ x
2
+ x
3
)
1
+
T`ım to
.
adˆo
.
cu
’
a vecto
.
x =3e
1
− e
2
=(3, −1, 0) dˆo

´
iv´o
.
ico
.
so
.
’
(E
1
, E
2
, E
3
).
2
+
Ch´u
.
ng minh r˘a
`
ng L l`a ph´ep bdtt.