Bài tập toán cao cấp Tập 1 part 7 pps

4.3. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh thuˆa
`
n nhˆa
´
t 167
cu
’
ahˆe
.
phu
.
o
.
ng tr`ınh (4.10) d
u
.
o
.
.
cgo
.
il`ahˆe
.

nghiˆe
.
mco
.
ba
’
n cu
’
an´onˆe
´
u
mˆo
˜
i nghiˆe
.
mcu
’
ahˆe
.
(4.10) d
ˆe
`
u l`a tˆo
’
ho
.
.
p tuyˆe
´
nt´ınh cu

’
a c´ac nghiˆe
.
m
e
1
,e
2
, ,e
m
.
D
-
i
.
nh l´y (vˆe
`
su
.
.
tˆo
`
nta
.
ihˆe
.
nghiˆe
.
mco
.

ba
’
n). Nˆe
´
uha
.
ng cu
’
a ma trˆa
.
n
cu
’
ahˆe
.
(4.10) b´eho
.
nsˆo
´
ˆa
’
nth`ıhˆe
.
(4.10) c´o hˆe
.
nghiˆe
.
mco
.
ba

’
n.
168 Chu
.
o
.
ng 4. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh
Phu
.
o
.
ng ph´ap t`ım hˆe
.
nghiˆe
.
mco
.
ba
’
n
1) D
ˆa

`
u tiˆen cˆa
`
n t´ach ra hˆe
.
ˆa
’
nco
.
so
.
’
(gia
’
su
.
’
d
´ol`ax
1
, ,x
r
)v`athu
du
.
o
.
.
chˆe
.

a
11
x
1
+ ···+ a
1r
x
r
= −a
1r+1
x
r+1
−···−a
1n
x
n
,

a
r1
x
1
+ ···+ a
rr
x
r
= −a
rr+1
x
r+1

−···−a
rn
x
n
.





(4.12)
2) Gia
’
su
.
’
hˆe
.
(4.12) c´o nghiˆe
.
ml`a
x
i
=

α
(i)
1
,α
(i)

2
, ,α
(i)
r
; x
r+1
, ,x
n
)

; i = 1,r.
Cho c´ac ˆa
’
ntu
.
.
do c´ac gi´a tri
.
x
r+1
=1,x
r+2
=0, ,x
n
=0
ta thu du
.
o
.
.

c
e
1
=

α
(1)
1
,α
(1)
2
, ,α
(1)
r
;1, 0, ,0

Tu
.
o
.
ng tu
.
.
,v´o
.
i x
r+1
=0,x
r+2
=1,x

r+3
=0, ,x
n
= 0 ta c´o
e
2
=

α
(2)
1
, ,α
(2)
r
;0, 1, 0, ,0

,
v`a sau c`ung v´o
.
i x
r+1
=0, ,x
n−1
=0,x
n
=1tathudu
.
o
.
.

c
e
k
=(α
(k)
1
, ,α
(k)
r
, 0, ,1),k= n − r.
Hˆe
.
c´ac nghiˆe
.
m e
1
,e
2
, ,e
k
v`u
.
athudu
.
o
.
.
cl`ahˆe
.
nghiˆe

.
mco
.
ba
’
n.
C
´
AC V
´
IDU
.
V´ı d u
.
1. T`ım nghiˆe
.
mtˆo
’
ng qu´at v`a hˆe
.
nghiˆe
.
mco
.
ba
’
ncu
’
ahˆe
.

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

.
ng tr`ınh tuyˆe
´
n t´ınh thuˆa
`
n nhˆa
´
t 169
Gia
’
i. 1) V`ı sˆo
´
phu
.
o
.
ng tr`ınh b´eho
.
nsˆo
´
ˆa
’
n nˆen tˆa
.
pho
.
.
p nghiˆe
.
mcu

’
a
hˆe
.
l`a vˆo ha
.
n.
Hiˆe
’
n nhiˆen ha
.
ng cu
’
a ma trˆa
.
ncu
’
ahˆe
.
b˘a
`
ng2v`ı trong c´ac d
i
.
nh th´u
.
c
con cˆa
´
p2c´odi

.
nh th´u
.
c con





2 −1
41





=0.
Do vˆa
.
yhˆe
.
d
˜a cho tu
.
o
.
ng d
u
.
o

.
ng v´o
.
ihˆe
.
2x
1
− x
3
= −x
1
− x
4
,
4x
1
+ x
3
= −2x
2
+3x
4
.
T`u
.
d´o suy ra
x
1
=
−3x

2
+2x
4
6
,x
3
=
5
3
x
4
. (4.13)
Do d
´otˆa
.
pho
.
.
p nghiˆe
.
mcu
’
ahˆe
.
c´o da
.
ng

−3α +2β
6

; α;
5
3
β; β


∀α, β ∈ R

(*)
2) Nˆe
´
u trong (4.13) ta cho c´ac ˆa
’
ntu
.
.
do bo
.
’
i c´ac gi´a tri
.
lˆa
`
nlu
.
o
.
.
t
b˘a

`
ng c´ac phˆa
`
ntu
.
’
cu
’
a c´ac cˆo
.
tdi
.
nh th´u
.
c





10
01





(=0)
th`ı thu d
u

.
o
.
.
c c´ac nghiˆe
.
m
e
1
=

−
1
2
;1;0;0

v`a e
2
=

1
3
;0;
5
3
;1

.
D
´ol`ahˆe

.
nghiˆe
.
mco
.
ba
’
ncu
’
ahˆe
.
phu
.
o
.
ng tr`ınh d˜a cho v`a nghiˆe
.
mtˆo
’
ng
qu´at cu
’
ahˆe
.
d
˜a cho c´o thˆe
’
biˆe
’
udiˆe

˜
ndu
.
´o
.
ida
.
ng
X = λe
1
+ µe
2
= λ

−
1
2
;1;0;0

+ µ

1
3
;0;
5
3
;1

170 Chu
.

o
.
ng 4. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh
trong d´o λ v`a µ l`a c´ac h˘a
`
ng sˆo
´
t`uy ´y:
X =

−3λ +2µ
6
; λ;
5
3
µ; µ


∀λ, µ ∈ R

.
Khi cho λ v`a µ c´ac gi´a tri

.
sˆo
´
kh´ac nhau ta s˜e thu d
u
.
o
.
.
c c´ac nghiˆe
.
m
riˆeng kh´ac nhau. 
V´ı d u
.
2. Gia
’
ihˆe
.
x
1
+2x
2
− x
3
=0,
−3x
1
− 6x
2

+3x
3
=0,
7x
1
+14x
2
− 7x
3
=0.





Gia
’
i. Hˆe
.
d
˜a cho tu
.
o
.
ng d
u
.
o
.
ng v´o

.
iphu
.
o
.
ng tr`ınh
x
1
+2x
2
− x
3
=0.
T`u
.
d
´o suy ra nghiˆe
.
mcu
’
ahˆe
.
l`a:
x
1
= −2x
2
+ x
3
,

x
2
= x
2
,
x
3
= x
3
; x
2
v`a x
3
t`uy ´y,
hay du
.
´o
.
ida
.
ng kh´ac
e =(−2x
2
+ x
3
; x
2
; x
3
).

Cho x
2
=1,x
3
= 0 ta c´o
e
1
=(−2; 1; 0),
la
.
ichox
2
=0,x
3
=1tathudu
.
o
.
.
c
e
2
=(1, 0, 1).
Hai h`ang e
1
v`a e
2
l`a dˆo
.
clˆa

.
p tuyˆe
´
n t´ınh v`a mo
.
i nghiˆe
.
mcu
’
ahˆe
.
dˆe
`
uc´o
da
.
ng
X = λe
1
+ µe
2
=(−2λ + µ; λ; µ)
4.3. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´

n t´ınh thuˆa
`
n nhˆa
´
t 171
trong d´o λ v`a µ l`a c´ac sˆo
´
t`uy ´y. 
V´ı du
.
3. T`ım nghiˆe
.
mtˆo
’
ng qu´at v`a hˆe
.
nghiˆe
.
mco
.
ba
’
ncu
’
ahˆe
.
phu
.
o
.

ng
tr`ınh
x
1
+3x
2
+3x
3
+2x
4
+4x
5
=0,
x
1
+4x
2
+5x
3
+3x
4
+7x
5
=0,
2x
1
+5x
2
+4x
3

+ x
4
+5x
5
=0,
x
1
+5x
2
+7x
3
+6x
4
+10x
5
=0.









Gia
’
i. B˘a
`
ng c´ac ph´ep biˆe

´
nd
ˆo
’
iso
.
cˆa
´
p, dˆe
˜
d`ang thˆa
´
yr˘a
`
ng hˆe
.
d˜acho
c´o thˆe
’
d
u
.
avˆe
`
hˆe
.
bˆa
.
c thang sau d
ˆay

x
1
+3x
2
+3x
3
+2x
4
+4x
5
=0,
x
2
+2x
3
+ x
4
+3x
5
=0,
x
4
=0.





Ta s˜e cho
.

n x
1
, x
2
v`a x
4
l`am ˆa
’
nco
.
so
.
’
; c`on x
3
v`a x
5
l`am ˆa
’
ntu
.
.
do. Ta
c´o hˆe
.
x
1
+3x
2
+2x

4
= −3x
3
−4x
5
,
x
2
+ x
4
= −2x
3
−3x
5
,
x
4
=0.





Gia
’
ihˆe
.
n`ay ta thu d
u
.

o
.
.
c nghiˆe
.
mtˆo
’
ng qu´at l`a
x
1
=3x
3
+5x
5
,
x
2
= −2x
3
−3x
5
,
x
4
=0.
Cho c´ac ˆa
’
ntu
.
.

do lˆa
`
nlu
.
o
.
.
t c´ac gi´a tri
.
b˘a
`
ng x
3
=1,x
5
= 0 (khi d´o
x
1
=3,x
2
=2,x
3
=1,x
4
=0,x
5
= 0) v`a cho x
3
=0,x
5

= 1 (khi d´o
x
1
=5,x
2
=3,x
3
=0,x
4
=0,x
5
= 1) ta thu du
.
o
.
.
chˆe
.
nghiˆe
.
mco
.
ba
’
n
e
1
= (3; −2; 1; 0; 0),
e
2

= (5; −3; 0; 0; 1).
172 Chu
.
o
.
ng 4. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh
T`u
.
d
´o nghiˆe
.
mtˆo
’
ng qu´at c´o thˆe
’
viˆe
´
tdu
.
´o
.
ida

.
ng
X = λ(3; −2; 1; 0; 0) + µ(5; −3; 0; 0; 1)
=(3λ +5µ; −2λ −3µ; λ;0;µ); ∀λ, µ ∈ R.
B˘a
`
ng c´ach cho λ v`a µ nh˜u
.
ng gi´a tri
.
sˆo
´
kh´ac nhau ta thu du
.
o
.
.
c c´ac
nghiˆe
.
m riˆeng kh´ac nhau. D
ˆo
`
ng th`o
.
i, mo
.
i nghiˆe
.
m riˆeng c´o thˆe

’
thu
d
u
.
o
.
.
ct`u
.
d
´ob˘a
`
ng c´ach cho
.
n c´ac hˆe
.
sˆo
´
λ v`a µ th´ıch ho
.
.
p. 
4.3. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe

´
n t´ınh thuˆa
`
n nhˆa
´
t 173
B
`
AI T
ˆ
A
.
P
Gia
’
ic´achˆe
.
phu
.
o
.
ng tr`ınh thuˆa
`
n nhˆa
´
t
1.
x
1
+2x

2
+3x
3
=0,
2x
1
+3x
2
+4x
3
=0,
3x
1
+4x
2
+5x
3
=0.





.
(D
S. x
1
= α, x
2
= −2α, x

3
= α, ∀α ∈ R)
2.
x
1
+ x
2
+ x
3
=0,
3x
1
− x
2
− x
3
=0,
2x
1
+3x
2
+ x
3
=0.





.(D

S. x
1
= x
2
= x
3
=0)
3.
3x
1
− 4x
2
+ x
3
−x
4
=0,
6x
1
− 8x
2
+2x
3
+3x
4
=0.

(DS. x
1
=

4α −β
3
, x
2
= α, x
3
= β, x
4
=0;α, β ∈ R t`uy ´y)
4.
3x
1
+2x
2
− 8x
3
+6x
4
=0,
x
1
− x
2
+4x
3
− 3x
4
=0.

(DS. x

1
=0,x
2
= α, x
3
= β, x
4
=
−α +4β
3
; α, β ∈ R t`uy ´y)
5.
x
1
− 2x
2
+3x
3
− x
4
=0,
x
1
+ x
2
− x
3
+2x
4
=0,

4x
1
− 5x
2
+8x
3
+ x
4
=0.





(D
S. x
1
= −
1
4
α, x
2
= α, x
3
=
3
4
α, x
4
=0;α ∈ R t`uy ´y)

6.
3x
1
− x
2
+2x
3
+ x
4
=0,
x
1
+ x
2
−x
3
− x
4
=0,
5x
1
+ x
2
− x
3
=0.






(D
S. x
1
= −
α
4
, x
2
=
5α
4
+ β, x
3
= α, x
4
= β; α, β ∈ R t`uy ´y)
7.
2x
1
+ x
2
+ x
3
=0,
3x
1
+2x
2
− 3x

3
=0,
x
1
+3x
2
− 4x
3
=0,
5x
1
+ x
2
− 2x
3
=0.









174 Chu
.
o
.
ng 4. Hˆe

.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh
(DS. x
1
=
α
7
, x
2
=
9α
7
, x
3
= α; α ∈ R t`uy ´y)
4.3. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh thuˆa

`
n nhˆa
´
t 175
T`ım nghiˆe
.
mtˆo
’
ng qu´at v`a hˆe
.
nghiˆe
.
mco
.
ba
’
ncu
’
a c´ac hˆe
.
phu
.
o
.
ng
tr`ınh
8.
9x
1
+21x

2
− 15x
3
+5x
4
=0,
12x
1
+28x
2
− 20x
3
+7x
4
=0.

(D
S. Nghiˆe
.
mtˆo
’
ng qu´at: x
1
= −
7
3
x
2
+
5

3
x
3
, x
4
=0.
Hˆe
.
nghiˆe
.
mco
.
ba
’
n e
1
=(−7, 3, 0, 0), e
2
=(5, 0, 3, 0))
9.
14x
1
+35x
2
−7x
3
− 63x
4
=0,
−10x

1
− 25x
2
+5x
3
+45x
4
=0,
26x
1
+65x
2
− 13x
3
− 117x
4
=0.





(D
S. Nghiˆe
.
mtˆo
’
ng qu´at: x
3
=2x

1
+5x
2
− 9x
3
.
Hˆe
.
nghiˆe
.
mco
.
ba
’
n: e
1
=(1, 0, 2, 0); e
2
=(0, 1, 5, 0); e
3
=
(0, 0, −9, 1))
10.
x
1
+4x
2
+2x
3
− 3x

5
=0,
2x
1
+9x
2
+5x
3
+2x
4
+ x
5
=0,
x
1
+3x
2
+ x
3
− 2x
4
−9x
5
=0.





(D

S. Nghiˆe
.
mtˆo
’
ng qu´at: x
1
=2x
3
+8x
4
, x
2
= −x
2
− 2x
4
; x
5
=0.
Hˆe
.
nghiˆe
.
mco
.
ba
’
n: e
1
=(2, −1, 1, 0, 0); e

2
=(8, −2, 0, 1, 0)
11.
x
1
+2x
2
+4x
3
− 3x
4
=0,
3x
1
+5x
2
+6x
3
−4x
4
=0,
4x
1
+5x
2
− 2x
3
+3x
4
=0,

3x
1
+8x
2
+24x
3
−19x
4
=0.









(D
S. Nghiˆe
.
mtˆo
’
ng qu´at: x
1
=8x
3
− 7x
4
, x

2
= −6x
3
+5x
4
.
Hˆe
.
nghiˆe
.
mco
.
ba
’
n: e
1
=(8, −6, 1, 0), e
2
=(−7, 5, 0, 1))
12.
x
1
+2x
2
− 2x
3
+ x
4
=0,
2x

1
+4x
2
+2x
3
− x
4
=0,
x
1
+2x
2
+4x
3
− 2x
4
=0,
4x
1
+8x
2
− 2x
3
+ x
4
=0.










(D
S. Nghiˆe
.
mtˆo
’
ng qu´at x
1
= −2x
2
, x
4
=2x
3
.
Hˆe
.
nghiˆe
.
mco
.
ba
’
n: e
1
=(−2, 1, 0, 0), e

2
=(0, 0, 1, 2))
176 Chu
.
o
.
ng 4. Hˆe
.
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh
13.
x
1
+2x
2
+3x
3
+4x
4
+5x
5
=0,
2x
1
+3x

2
+4x
3
+5x
4
+ x
5
=0,
3x
1
+4x
2
+5x
3
+ x
4
+2x
5
=0,
x
1
+3x
2
+5x
3
+12x
4
+9x
5
=0,

4x
1
+5x
2
+6x
3
− 3x
4
+3x
5
=0.















(D
S. Nghiˆe
.
mtˆo

’
ng qu´at x
1
= x
3
+15x
5
, x
2
= −2x
3
−12x
5
, x
4
= x
5
.
Hˆe
.
nghiˆe
.
mco
.
ba
’
n: e
1
=(1, −2, 1, 0, 0), e
2

= (15, −12, 0, 1, 1))
Chu
.
o
.
ng 5
Khˆong gian Euclide
R
n
5.1 D
-
i
.
nh ngh˜ıa khˆong gian n-chiˆe
`
uv`amˆo
.
tsˆo
´
kh´ai niˆe
.
mco
.
ba
’
nvˆe
`
vecto
.
177

5.2 Co
.
so
.
’
.D
-
ˆo
’
ico
.
so
.
’
188
5.3 Khˆong gian vecto
.
Euclid. Co
.
so
.
’
tru
.
.
c chuˆa
’
n201
5.4 Ph´ep biˆe
´

nd
ˆo
’
i tuyˆe
´
nt´ınh 213
5.4.1 D
-
i
.
nhngh˜ıa 213
5.4.2 Ma trˆa
.
ncu
’
aph´epbdtt 213
5.4.3 C´ac ph´ep to´an . . . . . . . . . . . . . . . . 215
5.4.4 Vecto
.
riˆeng v`a gi´a tri
.
riˆeng . . . . . . . . . 216
5.1 D
-
i
.
nh ngh˜ıa khˆong gian n-chiˆe
`
uv`a
mˆo

.
tsˆo
´
kh´ai niˆe
.
mco
.
ba
’
nvˆe
`
vecto
.
1
◦
. Gia
’
su
.
’
n ∈ N.Tˆa
.
pho
.
.
pmo
.
ibˆo
.
c´o thˆe

’
c´o (x
1
,x
2
, ,x
n
)gˆo
`
m n
sˆo
´
thu
.
.
c (ph´u
.
c) du
.
o
.
.
cgo
.
il`akhˆong gian thu
.
.
c (ph´u
.
c) n-chiˆe

`
u v`a du
.
o
.
.
c
178 Chu
.
o
.
ng 5. Khˆong gian Euclide R
n
k´yhiˆe
.
ul`aR
n
(C
n
). Mˆo
˜
ibˆo
.
sˆo
´
d´odu
.
o
.
.

cchı
’
bo
.
’
i
x =(x
1
,x
2
, ,x
n
)
v`a du
.
o
.
.
cgo
.
il`adiˆe
’
m hay vecto
.
cu
’
a R
n
(C
n

). C´ac sˆo
´
x
1
, ,x
n
du
.
o
.
.
c
go
.
il`ato
.
adˆo
.
cu
’
adiˆe
’
m (cu
’
a vecto
.
) x hay c´ac th`anh phˆa
`
ncu
’

a vecto
.
x.
Hai vecto
.
x =(x
1
, ,x
n
)v`ay =(y
1
, ,y
n
)cu
’
a R
n
du
.
o
.
.
c xem l`a
b˘a
`
ng nhau nˆe
´
u c´ac to
.
ad

ˆo
.
tu
.
o
.
ng ´u
.
ng cu
’
ach´ung b˘a
`
ng nhau
x
i
= y
i
∀i = 1,n.
C´ac vecto
.
x =(x
1
, ,x
n
), y =(y
1
, ,y
n
) c´o thˆe
’

cˆo
.
ng v´o
.
i nhau
v`a c´o thˆe
’
nhˆan v´o
.
i c´ac sˆo
´
α,β, l`a sˆo
´
thu
.
.
cnˆe
´
u khˆong gian d
u
.
o
.
.
cx´et
l`a khˆong gian thu
.
.
cv`al`asˆo
´

ph´u
.
cnˆe
´
u khˆong gian du
.
o
.
.
cx´et l`a khˆong
gian ph´u
.
c.
Theo di
.
nh ngh˜ıa: 1
+
tˆo
’
ng cu
’
a vecto
.
x v`a y l`a vecto
.
x + y
def
=(x
1
+ y

1
,x
2
+ y
2
, ,x
n
+ y
n
). (5.1)
2
+
t´ıch cu
’
a vecto
.
x v´o
.
isˆo
´
α hay t´ıch sˆo
´
α v´o
.
i vecto
.
x l`a vecto
.
αx = xα
def

=(αx
1
,αx
2
, ,αx
n
). (5.2)
Hai ph´ep to´an 1
+
v`a 2
+
tho
’
a m˜an c´ac t´ınh chˆa
´
t (tiˆen dˆe
`
) sau dˆay
I. x + y = y + x, ∀x,y ∈ R
n
(C
n
),
II. (x + y)+z = x +(y + z) ∀x, y, z ∈= R
n
(C
n
),
III. Tˆo
`

nta
.
i vecto
.
- khˆong θ =(0, 0, ,0

 
n
) ∈ R
n
sao cho
x + θ = θ + x = x,
IV. Tˆo
`
nta
.
i vecto
.
dˆo
´
i −x =(−1)x =(−x
1
, −x
2
, ,−x
n
) sao cho
x +(−x)=θ,
V. 1 · x = x,
5.1. D

-
i
.
nh ngh˜ıa khˆong gian n-chiˆe
`
uv`amˆo
.
tsˆo
´
kh´ai niˆe
.
mco
.
ba
’
nvˆe
`
vecto
.
179
VI. α(βx)=(αβ)x, α, β ∈ R (C),
VII. (α + β)x = αx + βx,
VIII. α(x + y)=αx + αy
trong d´o α v`a β l`a c´ac sˆo
´
, c`on x, y ∈ R
n
(C
n
).

D
-
i
.
nh ngh˜ıa 5.1.1. 1
+
Gia
’
su
.
’
V l`a tˆa
.
pho
.
.
p khˆong rˆo
˜
ng t `uy ´y v´o
.
i c´ac
phˆa
`
ntu
.
’
d
u
.
o

.
.
ck´yhiˆe
.
ul`ax,y,z, Tˆa
.
pho
.
.
p V d
u
.
o
.
.
cgo
.
i l`a khˆong gian
tuyˆe
´
n t´ınh (hay khˆong gian vecto
.
) nˆe
´
u ∀x, y ∈Vx´ac di
.
nh du
.
o
.

.
c phˆa
`
n
tu
.
’
x + y ∈V(go
.
i l`a tˆo
’
ng cu
’
a x v`a y)v`a∀α ∈ R (C)v`a∀x ∈Vx´ac
di
.
nh du
.
o
.
.
c phˆa
`
ntu
.
’
αx ∈V(go
.
i l`a t´ıch cu
’

asˆo
´
α v´o
.
i phˆa
`
ntu
.
’
x) sao
cho c´ac tiˆen d
ˆe
`
I-VI II du
.
o
.
.
c tho
’
a m˜an.
Khˆong gian tuyˆe
´
n t´ınh v´o
.
i ph´ep nhˆan c´ac phˆa
`
ntu
.
’

cu
’
an´ov´o
.
i c´ac
sˆo
´
thu
.
.
c (ph´u
.
c) du
.
o
.
.
cgo
.
i l`a khˆong gian tuyˆe
´
n t´ınh thu
.
.
c (tu
.
o
.
ng ´u
.

ng:
ph´u
.
c).
Khˆong gian R
n
c´o thˆe
’
xem nhu
.
mˆo
.
tv´ıdu
.
vˆe
`
khˆong gian tuyˆe
´
n
t´ınh, c´ac v´ı du
.
kh´ac s˜e d
u
.
o
.
.
cx´et vˆe
`
sau. V`a trong gi´ao tr`ınh n`ay ta

luˆon gia
’
thiˆe
´
tr˘a
`
ng c´ac khˆong gian d
u
.
o
.
.
cx´et l`a nh˜u
.
ng khˆong gian thu
.
.
c.
2
◦
. Cho hˆe
.
gˆo
`
m m vecto
.
n-chiˆe
`
u
x

1
,x
2
, ,x
m
. (5.3)
Khi d
´o vecto
.
da
.
ng
y = α
1
x
1
+ α
2
x
2
+ ···+ α
m
x
m
; α
1
,α
2
, ,α
m

∈ R.
d
u
.
o
.
.
cgo
.
il`atˆo
’
ho
.
.
p tuyˆe
´
nt´ınh cu
’
a c´ac vecto
.
d˜a cho hay vecto
.
y biˆe
’
u
diˆe
˜
n tuyˆe
´
n t´ınh du

.
o
.
.
c qua c´ac vecto
.
(5.3).
D
-
i
.
nh ngh˜ıa 5.1.2. 1
+
Hˆe
.
vecto
.
(5.3) du
.
o
.
.
cgo
.
il`ahˆe
.
dˆo
.
clˆa
.

p tuyˆe
´
n
t´ınh (d
ltt) nˆe
´
ut`u
.
d
˘a
’
ng th´u
.
c vecto
.
λ
1
x
1
+ λ
2
x
2
+ ···+ λ
m
x
m
= θ (5.4)
k´eo theo λ
1

= λ
2
= ···= λ
m
=0.
180 Chu
.
o
.
ng 5. Khˆong gian Euclide R
n
2
+
Hˆe
.
(5.3) go
.
il`ahˆe
.
phu
.
thuˆo
.
c tuyˆe
´
n t´ınh (pttt) nˆe
´
utˆo
`
nta

.
i c´ac sˆo
´
λ
1
,λ
2
, ,λ
m
khˆong dˆo
`
ng th`o
.
ib˘a
`
ng 0 sao cho d˘a
’
ng th´u
.
c (5.4) du
.
o
.
.
c
tho
’
a m˜an.
Sˆo
´

nguyˆen du
.
o
.
ng r du
.
o
.
.
cgo
.
il`aha
.
ng cu
’
ahˆe
.
vecto
.
(5.3) nˆe
´
u
a) C´o mˆo
.
ttˆa
.
pho
.
.
p con gˆo

`
m r vecto
.
cu
’
ahˆe
.
(5.3) lˆa
.
p th`anh hˆe
.
d
ltt.
b) Mo
.
itˆa
.
p con gˆo
`
m nhiˆe
`
uho
.
n r vecto
.
cu
’
ahˆe
.
(5.3) dˆe

`
u phu
.
thuˆo
.
c
tuyˆe
´
n t´ınh.
Dˆe
’
t`ım ha
.
ng cu
’
ahˆe
.
vecto
.
ta lˆa
.
p ma trˆa
.
n c´ac to
.
adˆo
.
cu
’
an´o

A =






a
11
a
12
a
1n
a
21
a
22
a
2n
.
.
.
.
.
.
.
.
.
.
.

.
a
m1
a
m2
a
mn






D
-
i
.
nh l´y. Ha
.
ng cu
’
ahˆe
.
vecto
.
(5.3) b˘a
`
ng ha
.
ng cu

’
a ma trˆa
.
n A c´ac to
.
a
d
ˆo
.
cu
’
a n´o.
T`u
.
d´o, dˆe
’
kˆe
´
t luˆa
.
nhˆe
.
vecto
.
(5.3) dltt hay pttt ta cˆa
`
nlˆa
.
p ma trˆa
.

n
to
.
adˆo
.
A cu
’
ach´ung v`a t´ınh r(A):
1) Nˆe
´
u r(A)=m th`ı hˆe
.
(5.3) dˆo
.
clˆa
.
p tuyˆe
´
n t´ınh.
2) Nˆe
´
u r(A)=s<mth`ı hˆe
.
(5.3) phu
.
thuˆo
.
c tuyˆe
´
n t´ınh.

C
´
AC V
´
IDU
.
V´ı d u
.
1. Ch´u
.
ng minh r˘a
`
ng hˆe
.
vecto
.
a
1
,a
2
, ,a
m
(m>1) phu
.
thuˆo
.
c
tuyˆe
´
n t´ınh khi v`a chı

’
khi ´ıt nhˆa
´
tmˆo
.
t trong c´ac vecto
.
cu
’
ahˆe
.
l`a tˆo
’
ho
.
.
p
tuyˆe
´
n t´ınh cu
’
a c´ac vecto
.
c`on la
.
i.
Gia
’
i. 1
+

Gia
’
su
.
’
hˆe
.
a
1
,a
2
, ,a
m
phu
.
thuˆo
.
c tuyˆe
´
n t´ınh. Khi d´o
tˆo
`
nta
.
i c´ac sˆo
´
α
1
,α
2

, ,α
m
khˆong dˆo
`
ng th`o
.
ib˘a
`
ng 0 sao cho
α
1
a
1
+ α
2
a
2
+ ···+ α
m
a
m
= θ.
Gia
’
su
.
’
α
m
= 0. Khi d´o

a
m
= β
1
a
1
+ β
2
a
2
+ ···+ β
m−1
a
m−1
,β
i
=
α
i
α
m
5.1. D
-
i
.
nh ngh˜ıa khˆong gian n-chiˆe
`
uv`amˆo
.
tsˆo

´
kh´ai niˆe
.
mco
.
ba
’
nvˆe
`
vecto
.
181
t´u
.
cl`aa
m
biˆe
’
udiˆe
˜
n tuyˆe
´
n t´ınh qua c´ac vecto
.
c`on la
.
i.
2
+
Ngu

.
o
.
.
cla
.
i, ch˘a
’
ng ha
.
nnˆe
´
u vecto
.
a
m
biˆe
’
udiˆe
˜
n tuyˆe
´
n t´ınh qua
a
1
,a
2
, ,a
m−1
a

m
= β
1
a
1
+ β
2
a
2
+ ···+ β
m−1
a
m−1
th`ı ta c´o
β
1
a
1
+ β
2
a
2
+ ···+ β
m−1
a
m−1
+(−1)a
m
= θ.
Do d´ohˆe

.
d˜a cho phu
.
thuˆo
.
c tuyˆe
´
n t´ınh v`ı trong d˘a
’
ng th´u
.
ctrˆenc´ohˆe
.
sˆo
´
cu
’
a a
m
l`a kh´ac 0 (cu
.
thˆe
’
l`a = −1). 
V´ı d u
.
2. Ch´u
.
ng minh r˘a
`

ng mo
.
ihˆe
.
vecto
.
c´o ch´u
.
a vecto
.
-khˆong l`a hˆe
.
phu
.
thuˆo
.
c tuyˆe
´
n t´ınh.
Gia
’
i. Vecto
.
- khˆong luˆon luˆon biˆe
’
udiˆe
˜
ndu
.
o

.
.
cdu
.
´o
.
ida
.
ng tˆo
’
ho
.
.
p
tuyˆe
´
n t´ınh cu
’
a c´ac vecto
.
a
1
,a
2
, ,a
m
:
θ =0· a
1
+0· a

2
+ ···+0· a
m
Do d´o theo di
.
nh ngh˜ıa hˆe
.
θ, a
1
, ,a
m
phu
.
thuˆo
.
c tuyˆe
´
n t´ınh (xem v´ı
du
.
1). 
V´ı d u
.
3. Ch´u
.
ng minh r˘a
`
ng mo
.
ihˆe

.
vecto
.
c´o ch´u
.
a hai vecto
.
b˘a
`
ng
nhau l`a hˆe
.
phu
.
thuˆo
.
c tuyˆe
´
n t´ınh.
Gia
’
i. Gia
’
su
.
’
trong hˆe
.
a
1

,a
2
, ,a
n
c´o hai vecto
.
a
1
= a
2
. Khi d´o
ta c´o thˆe
’
viˆe
´
t
a
1
=1· a
2
+0·a
3
+ ···+0· a
m
t´u
.
c l`a vecto
.
a
1

cu
’
ahˆe
.
c´o thˆe
’
biˆe
’
udiˆe
˜
ndu
.
´o
.
ida
.
ng tˆo
’
ho
.
.
p tuyˆe
´
n t´ınh
cu
’
a c´ac vecto
.
c`on la
.

i. Do d´o h ˆe
.
phu
.
thuˆo
.
c tuyˆe
´
n t´ınh (v´ı du
.
1). 
V´ı d u
.
4. Ch´u
.
ng minh r˘a
`
ng nˆe
´
uhˆe
.
m vecto
.
a
1
,a
2
, ,a
m
dˆo

.
clˆa
.
p
tuyˆe
´
n t´ınh th`ı mo
.
ihˆe
.
con cu
’
ahˆe
.
d
´oc˜ung dˆo
.
clˆa
.
p tuyˆe
´
n t´ınh.
Gia
’
i. Dˆe
’
cho x´ac di
.
nh ta x´et hˆe
.

con a
1
,a
2
, ,a
k
, k<mv`a ch´u
.
ng
minh r˘a
`
ng hˆe
.
con n`ay dˆo
.
clˆa
.
p tuyˆe
´
n t´ınh.
182 Chu
.
o
.
ng 5. Khˆong gian Euclide R
n
Gia
’
su
.

’
ngu
.
o
.
.
cla
.
i: hˆe
.
con a
1
,a
2
, ,a
k
phu
.
thuˆo
.
c tuyˆe
´
n t´ınh. Khi
d´o ta c´o c´ac d˘a
’
ng th´u
.
c vecto
.
α

1
a
1
+ α
2
a
2
+ ···+ α
k
a
k
= θ
trong d
´o c´o ´ıt nhˆa
´
tmˆo
.
t trong c´ac hˆe
.
sˆo
´
α
1
,α
2
, ,α
k
kh´ac 0. Ta viˆe
´
t

d
˘a
’
ng th´u
.
cd
´odu
.
´o
.
ida
.
ng
α
1
a
1
+ α
2
A
2
+ ···+ α
k
a
k
+ α
k+1
a
k+1
+ ···+ α

m
a
m
= θ
trong d´o ta gia
’
thiˆe
´
t α
k+1
=0, ,α
m
=0. D˘a
’
ng th´u
.
c sau c`ung n`ay
ch´u
.
ng to
’
hˆe
.
a
1
,a
2
, ,a
m
phu

.
thuˆo
.
c tuyˆe
´
n t´ınh. Mˆau thuˆa
˜
n. 
V´ı d u
.
5. Ch´u
.
ng minh r˘a
`
ng hˆe
.
vecto
.
cu
’
a khˆong gian R
n
e
1
=(1, 0, ,0),
e
2
=(0, 1, ,0),

e

n
=(0, ,0, 1)
l`a dˆo
.
clˆa
.
p tuyˆe
´
n t´ınh.
Gia
’
i. T`u
.
d˘a
’
ng th´u
.
c vecto
.
α
1
e
1
+ α
2
e
2
+ ···+ α
n
e

n
= θ
suy ra r˘a
`
ng
(α
1
,α
2
, ,α
n
)=(0, 0, ,0) ⇒ α
1
= α
2
= ···= α
n
=0.
v`a do d´ohˆe
.
e
1
,e
2
, ,e
n
dˆo
.
clˆa
.

p tuyˆe
´
n t´ınh. 
V´ı d u
.
6. Ch´u
.
ng minh r˘a
`
ng mo
.
ihˆe
.
gˆo
`
m n + 1 vecto
.
cu
’
a R
n
l`a hˆe
.
phu
.
thuˆo
.
c tuyˆe
´
n t´ınh.

Gia
’
i. Gia
’
su
.
’
n + 1 vecto
.
cu
’
ahˆe
.
l`a:
a
1
=(a
11
,a
21
, ,a
n1
)
a
2
=(a
12
,a
22
, ,a

n2
)

a
n+1
=(a
1,n+1
,a
2,n+1
, ,a
n,n+1
).
5.1. D
-
i
.
nh ngh˜ıa khˆong gian n-chiˆe
`
uv`amˆo
.
tsˆo
´
kh´ai niˆe
.
mco
.
ba
’
nvˆe
`

vecto
.
183
Khi d´ot`u
.
d˘a
’
ng th´u
.
c vecto
.
x
1
a
1
+ x
2
a
2
+ ···+ x
n
a
n
+ x
n+1
a
n+1
= θ
suy ra
a

11
x
1
+ a
12
x
2
+ ···+ a
1n+1
x
n+1
=0,

a
n1
x
1
+ a
n2
x
2
+ ···+ a
nn+1
x
n+1
=0.






D´ol`ahˆe
.
thuˆa
`
n nhˆa
´
t n phu
.
o
.
ng tr`ınh v´o
.
i(n +1) ˆa
’
n nˆen hˆe
.
c´o nghiˆe
.
m
khˆong tˆa
`
mthu
.
`o
.
ng v`a
(x
1
,x

2
, ,x
n
,x
n+1
) =(0, 0, ,0).
Do d
´o theo di
.
nh ngh˜ıa hˆe
.
d˜a x´et l`a phu
.
thuˆo
.
c tuyˆe
´
n t´ınh. 
V´ı d u
.
7. T`ım ha
.
ng cu
’
ahˆe
.
vecto
.
trong R
4

a
1
=(1, 1, 1, 1); a
2
=(1, 2, 3, 4);
a
3
=(2, 3, 2, 3); a
4
=(2, 4, 5, 6).
Gia
’
i. Ta lˆa
.
p ma trˆa
.
n c´ac to
.
ad
ˆo
.
v`a t`ım ha
.
ng cu
’
a n´o. Ta c´o
A =






1111
1234
2323
3456





h
2
− h
1
→ h

2
h
3
− 2h
1
→ h

3
h
4
− 3h
1
→ h


4
−→





1111
0123
0101
0123





h
3
− h
2
→ h

3
h
4
− h
2
→ h


4
→
−→





11 1 1
01 2 3
00−2 −3
00 0 0





.
T`u
.
d´o suy r˘a
`
ng r(A) = 3. Theo di
.
nh l´yd˜a nˆeu ha
.
ng cu
’
ahˆe
.

vecto
.
b˘a
`
ng 3. 
184 Chu
.
o
.
ng 5. Khˆong gian Euclide R
n
V´ı d u
.
8. Kha
’
o s´at su
.
.
phu
.
thuˆo
.
c tuyˆe
´
n t´ınh gi˜u
.
a c´ac vecto
.
cu
’

a R
4
:
a
1
=(1, 4, 1, 1); a
2
=(2, 3, −1, 1);
a
3
=(1, 9, 4, 2); a
4
=(1, −6, −5, −1).
Gia
’
i. Lˆa
.
p ma trˆa
.
n m`a c´ac h`ang cu
’
a n´o l`a c´ac vecto
.
d˜a cho v`a t`ım
ha
.
ng cu
’
an´o
S =






1411
23−11
1942
1 −6 −5 −1





⇒ r(A)=2.
Do d
´oha
.
ng cu
’
ahˆe
.
vecto
.
b˘a
`
ng 2. V`ı c´ac phˆa
`
ntu
.

’
cu
’
adi
.
nh th´u
.
c con
∆=





14
23





= −5 =0
n˘a
`
mo
.
’
hai h`ang dˆa
`
unˆena

1
v`a a
2
dˆo
.
clˆa
.
p tuyˆe
´
n t´ınh, c`on a
3
v`a a
4
biˆe
’
u
diˆe
˜
n tuyˆe
´
n t´ınh qua a
1
v`a a
2
. [Lu
.
u´yr˘a
`
ng mo
.

ic˘a
.
p vecto
.
cu
’
ahˆe
.
dˆe
`
u
d
ˆo
.
clˆa
.
p tuyˆe
´
n t´ınh v`ı ta c´o c´ac di
.
nh th´u
.
c con cˆa
´
p hai sau d
ˆay =0:






14
19





,





14
1 −6





,





23
19






,





23
1 −6





,





19
1 −6






.]
Ta t`ım c´ac biˆe
’
uth´u
.
cbiˆe
’
udiˆe
˜
n a
3
v`a a
4
qua a
1
v`a a
2
.
Ta viˆe
´
t
a
3
= ξ
1
a
1
+ ξ
2
a

2
hay l`a
(1, 9, 4, 2) = ξ
1
· (1, 4, 1, 1) + ξ
2
· (2, 3, −1, 1)
⇒ (1, 9, 4, 2) = (ξ
1
+2ξ
2
, 4ξ
1
+3ξ
2
,ξ
1
− ξ
2
,ξ
1
+ ξ
2
)
5.1. D
-
i
.
nh ngh˜ıa khˆong gian n-chiˆe
`

uv`amˆo
.
tsˆo
´
kh´ai niˆe
.
mco
.
ba
’
nvˆe
`
vecto
.
185
v`a thu du
.
o
.
.
chˆe
.
phu
.
o
.
ng tr`ınh
ξ
1
+2ξ

2
=1,
4ξ
1
+3ξ
2
=9,
ξ
1
− ξ
2
=4,
ξ
1
+ ξ
2
=2.









Ta ha
.
n chˆe
´

hai phu
.
o
.
ng tr`ınh d
ˆa
`
u. Di
.
nh th´u
.
ccu
’
ac´achˆe
.
sˆo
´
cu
’
a hai
phu
.
o
.
ng tr`ınh n`ay ch´ınh l`a d
i
.
nh th´u
.
c ∆ chuyˆe

’
nvi
.
.V`ı∆= 0 nˆen hˆe
.
hai phu
.
o
.
ng tr`ınh
ξ
1
+2ξ
2
=1
4ξ
1
+3ξ
2
=9
c´o nghiˆe
.
m duy nhˆa
´
tl`aξ
1
=3,ξ
2
= −1. Do d´o
a

3
=3a
1
− a
2
.
Tu
.
o
.
ng tu
.
.
ta c´o
a
4
=2a
2
− 3a
1
. 
B
`
AI T
ˆ
A
.
P
1. Ch´u
.

ng minh r˘a
`
ng trong khˆong gian R
3
:
1) Vecto
.
(x, y, z) l`a tˆo
’
ho
.
.
p tuyˆe
´
n t´ınh cu
’
a c´ac vecto
.
e
1
=(1, 0, 0),
e
2
=(0, 1, 0), e
3
=(0, 0, 1).
2) Vecto
.
x =(7, 2, 6) l`a tˆo
’

ho
.
.
p tuyˆe
´
n t´ınh cu
’
a c´ac vecto
.
a
1
=
(−3, 1, 2), a
2
=(−5, 2, 3), a
3
=(1, −1, 1).
2. H˜ay x´ac di
.
nh sˆo
´
λ dˆe
’
vecto
.
x ∈ R
3
l`a tˆo
’
ho

.
.
p tuyˆe
´
n t´ınh cu
’
a c´ac
vecto
.
a
1
,a
2
,a
3
∈ R
3
nˆe
´
u:
1) x =(1, 3, 5); a
1
=(3, 2, 5); a
2
=(2, 4, 7); a
3
=(5, 6,λ).
186 Chu
.
o

.
ng 5. Khˆong gian Euclide R
n
(DS. λ = 12)
2) x =(7, −2,λ); a
1
=(2, 3, 5); a
2
=(3, 7, 8); a
3
=(1, −6, 1).
(D
S. λ = 15)
3) x =(5, 9,λ); a
1
=(4, 4, 3); a
2
=(7, 2, 1); a
3
=(4, 1, 6).
(DS. ∀λ ∈ R)
3. Ch´u
.
ng minh r˘a
`
ng trong khˆong gian R
3
:
1) Hˆe
.

ba vecto
.
e
1
=(1, 0, 0), e
2
=(0, 1, 0), e
3
=(0, 0, 1) l`a hˆe
.
dltt.
2) Nˆe
´
u thˆem vecto
.
x ∈ R
3
bˆa
´
tk`y v`ao hˆe
.
th`ı hˆe
.
{e
1
,e
2
,e
3
,x}

l`a phu
.
thuˆo
.
c tuyˆe
´
n t´ınh.
3) Hˆe
.
gˆo
`
mbˆo
´
n vecto
.
bˆa
´
tk`ycu
’
a R
3
l`a pttt.
4. C´ac hˆe
.
vecto
.
sau dˆay trong khˆong gian R
3
l`a dltt hay pttt:
1) a

1
=(1, 2, 1); a
2
=(0, 1, 2); a
3
=(0, 0, 2). (DS. Dltt)
2) a
1
=(1, 1, 0); a
2
=(1, 0, 1); a
3
=(1, −2, 0). (DS. Dltt)
3) a
1
=(1, 3, 3); a
2
=(1, 1, 1); a
3
=(−2, −4, −4). (DS. Pttt)
4) a
1
=1, −3, 0); a
2
=(3, −3, 1); a
3
=(2, 0, 1). (DS. Pttt)
5) a
1
=(2, 3, 1); a

2
=(1, 1, 1); a
3
=(1, 2, 0). (DS. Pttt)
5. Gia
’
su
.
’
v
1
, v
2
v`a v
3
l`a hˆe
.
dˆo
.
clˆa
.
p tuyˆe
´
n t´ınh. Ch´u
.
ng minh r˘a
`
ng hˆe
.
sau d

ˆay c˜ung l`a dltt:
1) a
1
= v
1
+ v
2
; a
2
= v
1
+ v
3
; a
3
= v
1
− 2v
2
.
2) a
1
= v
1
+ v
3
; a
2
= v
3

− v
1
; a
3
= v
1
+ v
2
− v
3
.
6. Ch´u
.
ng minh r˘a
`
ng c´ac hˆe
.
vecto
.
sau d
ˆay l`a phu
.
thuˆo
.
c tuyˆe
´
n t´ınh.
Dˆo
´
iv´o

.
ihˆe
.
vecto
.
n`ao th`ı vecto
.
b l`a tˆo
’
ho
.
.
p tuyˆe
´
n t´ınh cu
’
a c´ac vecto
.
c`on la
.
i?
1) a
1
=(2, 0, −1), a
2
=(3, 0, −2), a
3
=(−1, 0, 1), b =(1, 2, 0).
(D
S. b khˆong l`a tˆo

’
ho
.
.
p tuyˆe
´
n t´ınh)
2) a
1
=(−2, 0, 1), a
2
=(1, −1, 0), a
3
=(0, 1, 2); b =(2, 3, 6).
(D
S. b l`a tˆo
’
ho
.
.
p tuyˆe
´
n t´ınh)
5.1. D
-
i
.
nh ngh˜ıa khˆong gian n-chiˆe
`
uv`amˆo

.
tsˆo
´
kh´ai niˆe
.
mco
.
ba
’
nvˆe
`
vecto
.
187
7. T`ım sˆo
´
cu
.
.
cda
.
i c´ac vecto
.
dltt trong c´ac hˆe
.
vecto
.
sau dˆay
1) a
1

=(2, 3, −1, 4); a
2
=(−1, 1, 2, 0); a
3
=(0, 0, 1, 1);
a
4
=(1, 4, 1, 4); a
5
=(2, 3, 0, 5). (DS. = 3)
2) a
1
=(1, 0, 0, 0); a
2
=(0, 1, 0, 0); a
3
=(0, 0, 1, 0)
a
4
=(0, 0, 0, 1); a
5
=(1, 2, 3, 4). (DS. = 4)
3) a
1
=(1, 1, 1, 1); a
2
=(1, 1, 1, 0); a
3
=(1, 1, 0, 0);
a

4
=(1, 0, 0, 0); a
5
=(1, 2, 3, 4). (DS. = 4)
Chı
’
dˆa
˜
n. Lˆa
.
p ma trˆa
.
n c´ac to
.
adˆo
.
m`a mˆo
˜
icˆo
.
tcu
’
an´ol`ato
.
adˆo
.
cu
’
a
vecto

.
cu
’
ahˆe
.
rˆo
`
i t´ınh ha
.
ng cu
’
a ma trˆa
.
n.
8. C´ac hˆe
.
vecto
.
sau dˆay trong khˆong gian R
4
l`a dltt hay pttt
1) a
1
=(1, 2, 3, 4), a
2
=(1, 2, 3, 4). (DS. Pttt)
2) a
1
=(1, 2, 3, 4), a
2

=(1, −2, −3, −4). (DS. Pttt)
3) a
1
=(1, 2, 3, 4), a
2
=(3, 6, 9, 12). (DS. Pttt)
4) a
1
=(1, 2, 3, 4), (a
2
=(1, 2, 3, 5). (DS. Dltt)
5) a
1
=(1, 0, 0, 0), a
2
=(0, 1, 0, 0), a
3
=(0, 0, 1, 0), a
4
=(0, 0, 0, 1)
v`a a l`a vecto
.
t`uy ´y cu
’
a R
4
.(DS. Pttt)
6) a
1
=(1, 1, 1, 1), a

2
=(0, 1, 1, 1), a
3
=(0, 0, 1, 1), a
4
=(0, 0, 0, 1).
(DS. Dltt)
7) a
1
=(1, 2, 3, 4), a
2
=(3, 6, 9, 12), a
3
=(1, 2, 3, 6). (DS. Pttt)
9. C´ac hˆe
.
vecto
.
sau dˆay dltt hay pttt. Trong tru
.
`o
.
ng ho
.
.
p pttt h˜ay chı
’
ra mˆo
.
tsu

.
.
pttt. H˜ay chı
’
ra mˆo
.
thˆe
.
con cu
.
.
cd
a
.
i n`ao d´ol`adltt.
1) a
1
=(2, 1, −2, −1), a
2
=(−9, 5, −6, 21), a
3
=(2, −5, −1, 3),
a
4
=(−1, −1, −1, 5), a
5
=(−1, 2, −3, 4).
(DS. a
1
+ a

2
+ a
3
− 3a
4
−2a
5
= θ; a
1
,a
2
,a
3
,a
4
)
2) a
1
=(1, 1, 1, 1), a
2
=(2, 0, 1, −1), a
3
=(3, −4, 0, −1),
a
4
= (13, −10, 3, −2). (DS. 2a
1
+ a
2
+3a

3
− a
4
= θ; a
1
,a
2
,a
3
)
3) a
1
=(1, −1, 1, −1), a
2
=(2, 0, 1, −1), a
3
=(3, −1, 1, −1),
a
4
=(4, −2, 1, −2). (DS. Hˆe
.
dˆo
.
clˆa
.
p tuyˆe
´
n t´ınh)
4) a
1

=(1, 2, −2, −1), a
2
=(−1, 0, 2, 1), a
3
=(0, 1, 0, 1),
a
4
=(3, 6, 0, 4). (DS. Hˆe
.
dˆo
.
clˆa
.
p tuyˆe
´
n t´ınh)
188 Chu
.
o
.
ng 5. Khˆong gian Euclide R
n
10. T´ınh ha
.
ng r cu
’
ahˆe
.
vecto
.

v`a chı
’
r˜o hˆe
.
d˜a cho l`a pttt hay dltt:
1) a
1
=(1, −2, 2, −8, 2), a
2
=(1, −2, 1, 5, 3), a
3
=(1, −2, 4, −7, 0).
(D
S. r =3,hˆe
.
dˆo
.
clˆa
.
p tuyˆe
´
n t´ınh)
2) a
1
=(2, 3, 1, −1), a
2
=(3, 1, 4, 2), a
3
=(1, 2, 3, −1),
a

4
=(1, −4, −7, 5). (DS. r =3,hˆe
.
pttt)
3) a
1
=(2, −1, −3, 2, −6), a
2
=(1, 5, −2, 3, 4), a
3
=(3, 4, −1, 5, 7),
a
4
=(3, −7, 4, 1, −7), a
5
=(0, 11, −5, 4, −4). (DS. r =3hˆe
.
pttt)
4) a
1
=(2, 1, 4, −4, 17), a
2
=(0, 0, 5, −7, 9), a
3
=
(2, 1, −6, 10, −11),
a
4
=(8, 4, 1, 5, 11), a
5

=(2, 2, 9, −11, 10). (DS. r =5,hˆe
.
dltt)
5.2 Co
.
so
.
’
.D
-
ˆo
’
ico
.
so
.
’
D
-
i
.
nh ngh˜ıa 5.2.1. Hˆe
.
vecto
.
E
1
,E
2
, ,E

n
gˆo
`
m n vecto
.
cu
’
a khˆong
gian vecto
.
R
n
du
.
o
.
.
cgo
.
il`amˆo
.
tco
.
so
.
’
cu
’
an´onˆe
´

u
1) hˆe
.
E
1
,E
2
, ,E
n
l`a hˆe
.
dltt;
2) mo
.
i vecto
.
x ∈ R
n
dˆe
`
ubiˆe
’
udiˆe
˜
n tuyˆe
´
n t´ınh du
.
o
.

.
c qua c´ac vecto
.
cu
’
ahˆe
.
E
1
, ,E
n
.
Ch´u´yr˘a
`
ng co
.
so
.
’
cu
’
a R
n
l`a mˆo
.
t hˆe
.
c´o th´u
.
tu

.
.
bˆa
´
tk`ygˆo
`
m n vecto
.
dˆo
.
clˆa
.
p tuyˆe
´
n t´ınh cu
’
a n´o.
D
iˆe
`
ukiˆe
.
n 2) c´o ngh˜ıa r˘a
`
ng ∀x ∈ R
n
, ∃(x
1
,x
2

, ,x
n
) sao cho
x = x
1
E
1
+ x
2
E
2
+ ···+ x
n
E
n
, (5.5)
trong d
´o x
1
,x
2
, ,x
n
l`a to
.
adˆo
.
cu
’
a vecto

.
x trong co
.
so
.
’
E
1
,E
2
, ,E
n
v`a (5.5) go
.
il`akhai triˆe
’
n vecto
.
x theo co
.
so
.
’
E
1
,E
2
, ,E
n
.

´
Y ngh˜ıa co
.
ba
’
ncu
’
a kh´ai niˆe
.
mco
.
so
.
’
l`a: c´ac ph´ep to´an tuyˆe
´
n t´ınh
trˆen c´ac vecto
.
trong co
.
so
.
’
cho tru
.
´o
.
c chuyˆe
’

n th`anh c´ac ph´ep to´an trˆen
c´ac sˆo
´
l`a to
.
adˆo
.
cu
’
ach´ung.
D
-
i
.
nh l´y 5.2.1. Trong khˆong gian R
n
:
1) To
.
ad
ˆo
.
cu
’
amˆo
.
t vecto
.
dˆo
´

iv´o
.
imˆo
.
tco
.
so
.
’
l`a duy nhˆa
´
t.
5.2. Co
.
so
.
’
.D
-
ˆo
’
ico
.
so
.
’
189
2) Mo
.
ihˆe

.
dltt gˆo
`
m n vecto
.
dˆe
`
ulˆa
.
p th`anh co
.
so
.
’
cu
’
a khˆong gian
R
n
.
Ta x´et vˆa
´
ndˆe
`
: Khi co
.
so
.
’
thay dˆo

’
i th`ı to
.
adˆo
.
cu
’
amˆo
.
t vecto
.
trong
khˆong gian R
n
thay dˆo
’
ithˆe
´
n`ao ?
Gia
’
su
.
’
trong khˆong gian R
n
c´o hai co
.
so
.

’
E :E
1
, E
2
, ,E
n
- “co
.
so
.
’
c˜u” (5.6)
E :E
1
,E
2
, ,E
n
- “co
.
so
.
’
m´o
.
i” (5.7)
V`ı E
1
,E

2
, ,E
n
∈ R
n
nˆen
E
1
= t
11
ε
1
+ t
21
ε
2
+ ···+ t
n1
ε
n
,
E
2
= t
12
ε
1
+ t
22
ε

2
+ ···+ t
n2
ε
n
,

E
n
= t
1n
ε
1
+ t
2n
ε
2
+ ···+ t
nn
ε
n
.










(5.8)
C´o thˆe
’
n´oi r˘a
`
ng co
.
so
.
’
E
1
, ,E
n
thu du
.
o
.
.
ct`u
.
co
.
so
.
’
E
1
, E

2
, ,E
n
nh`o
.
ma trˆa
.
n
T
EE
=






t
11
t
12
t
1n
t
21
t
22
t
2n
.

.
.
.
.
.
.
.
.
.
.
.
t
n1
t
n2
t
nn






(5.9)
trong d
´ocˆo
.
tth´u
.
i cu

’
a ma trˆa
.
n (5.9) ch´ınh l`a c´ac to
.
adˆo
.
cu
’
a vecto
.
E
i
trong co
.
so
.
’
(5.6).
Ma trˆa
.
n T = T
EE
trong (5.9) du
.
o
.
.
cgo
.

il`ama trˆa
.
n chuyˆe
’
n t`u
.
co
.
so
.
’
(5.6) d
ˆe
´
nco
.
so
.
’
(5.7). D
i
.
nh th´u
.
ccu
’
a ma trˆa
.
n chuyˆe
’

n detT =0
v`ı trong tru
.
`o
.
ng ho
.
.
p ngu
.
o
.
.
cla
.
i th`ı c´ac vecto
.
cˆo
.
t (v`a do d´o c´ac vecto
.
E
1
, ,E
n
) l`a phu
.
thuˆo
.
c tuyˆe

´
n t´ınh.
Nhu
.
vˆa
.
ydˆe
’
t`ım ma trˆa
.
n chuyˆe
’
nt`u
.
co
.
so
.
’
c˜u sang co
.
so
.
’
m´o
.
idˆa
`
u
tiˆen ta cˆa

`
n khai triˆe
’
n c´ac vecto
.
cu
’
aco
.
so
.
’
m´o
.
i theo co
.
so
.
’
c˜u. Tiˆe
´
pd
´o
ta lˆa
.
p ma trˆa
.
nm`acˆo
.
tth´u

.
i cu
’
a n´o l`a c´ac to
.
adˆo
.
cu
’
a vecto
.
th ´u
.
i cu
’
a
co
.
so
.
’
m´o
.
i trong co
.
so
.
’
c˜u. D´och´ınh l`a ma trˆa
.

n chuyˆe
’
n.
190 Chu
.
o
.
ng 5. Khˆong gian Euclide R
n
Gia
’
su
.
’
vecto
.
a ∈ R
n
v`a
a = x
1
ε
1
+ x
2
ε
2
+ ···+ x
n
ε

n
,
a = y
1
E
1
+ y
2
E
2
+ ···+ y
n
E
n
.
Khi d
´o quan hˆe
.
gi˜u
.
a c´ac to
.
adˆo
.
cu
’
ac`ung mˆo
.
t vecto
.

dˆo
´
iv´o
.
i hai co
.
so
.
’
kh´ac nhau (5.6) v`a (5.7) d
u
.
o
.
.
cmˆota
’
nhu
.
sau
x
1
= t
11
y
1
+ t
12
y
2

+ ···+ t
1n
y
n
,
x
2
= t
21
y
1
+ t
22
y
2
+ ···+ t
2n
y
n
,

x
n
= t
n1
y
1
+ t
n2
y

2
+ ···+ t
nn
y
n
.









(5.10)
hay l`a
X = T
EE
Y, (5.11)
X =






x
1
x

2
.
.
.
x
n






,Y=






y
1
y
2
.
.
.
y
n







T`u
.
d´oc˜ung suy ra
Y = T
−1
EE
X. (5.11*)
C
´
AC V
´
IDU
.
V´ı d u
.
1. Trong khˆong gian R
3
hˆe
.
c´ac vecto
.
E
1
(1, 0, 0), E
2
(0, 2, 0),

E
3
(0, 0, 3) l`a co
.
so
.
’
cu
’
a n´o.
Gia
’
i. 1) Hˆe
.
vecto
.
E
1
, E
2
, E
3
l`a hˆe
.
dˆo
.
clˆa
.
p tuyˆe
´

nt´ınh. Thˆa
.
tvˆa
.
y,
d
˘a
’
ng th´u
.
c vecto
.
α
1
E
1
+ α
2
E
2
+ α
3
E
3
=(0, 0, 0)
⇔ α
1
(1, 0, 0) + α
2
(0, 2, 0) + α

3
(0, 0, 3) = (0, 0, 0)
⇔ (α
1
, 2α
2
, 3α
3
)=(0, 0, 0)
⇔ α
1
= α
2
= α
3
=0.
5.2. Co
.
so
.
’
.D
-
ˆo
’
ico
.
so
.
’

191
2) Gia
’
su
.
’
x ∈ R
3
, x =(ξ
1
,ξ
2
,ξ
3
). Khi d´o
x = ξ
1
(1, 0, 0) +
ξ
2
2
(0, 2, 0) +
ξ
3
3
(0, 0, 3)
= ξ
1
E
1

+
ξ
2
2
E
2
+
ξ
3
3
E
3
t´u
.
cl`ax l`a tˆo
’
ho
.
.
p tuyˆe
´
n t´ınh cu
’
a E
1
, E
2
, E
3
. 

V´ı d u
.
2. Ch´u
.
ng minh r˘a
`
ng trong khˆong gian R
3
c´ac vecto
.
E
1
=
(2, 1, 1), E
2
=(1, 3, 1), E
3
=(−2, 1, 3) lˆa
.
p th`anh mˆo
.
tco
.
so
.
’
.T`ım to
.
a
d

ˆo
.
cu
’
a vecto
.
x =(−2, −4, 2) theo co
.
so
.
’
d´o.
Gia
’
i. 1) Hˆe
.
E
1
, E
2
, E
3
l`a dltt. Thˆa
.
tvˆa
.
y gia
’
su
.

’
α
1
E
1
+α
2
E
2
+α
3
E
3
=
θ ⇔
2α
1
+ α
2
−2α
3
=0,
α
1
+3α
2
+ α
3
=0,
α

1
+ α
2
+3α
3
=0.





Hˆe
.
n`ay c´o detA =0v`al`ahˆe
.
thuˆa
`
n nhˆa
´
tnˆen n´o chı
’
c´o nghiˆe
.
mtˆa
`
m
thu
.
`o
.

ng α
1
= α
2
= α
3
=0v`adod´o E
1
, E
2
, E
3
dˆo
.
clˆa
.
p tuyˆe
´
n t´ınh. Theo
di
.
nh l´y 1 (phˆa
`
n 2) c´ac vecto
.
n`ay lˆa
.
p th`anh co
.
so

.
’
cu
’
a R
3
.
2) Dˆe
’
khai triˆe
’
n vecto
.
x =(−2, −4, 2) theo co
.
so
.
’
E
1
, E
2
, E
3
ta d˘a
.
t
x = λ
1
E

1
+ λ
2
E
2
+ λ
3
E
3
v`a t`u
.
d´o
2λ
1
+ λ
2
− 2λ
3
= −2,
λ
1
+3λ
2
+ λ
3
= −4,
λ
1
+ λ
2

+3λ
3
=2.





Hˆe
.
n`ay c´o nghiˆe
.
ml`aλ
1
=1,λ
2
= −2, λ
3
= 1. Vˆa
.
y trong co
.
so
.
’
E
1
, E
2
, E

3
vecto
.
x c´o to
.
adˆo
.
l`a (1, −2, 1). 
V´ı d u
.
3. Ch´u
.
ng minh r˘a
`
ng ba vecto
.
E
1
=(1, 0, −2), E
2
=(−4, −1, 5),
E
3
=(1, 3, 4) lˆa
.
p th`anh co
.
so
.
’

cu
’
a R
3
.