Không gian vector

Chu
.
o
.
ng 3
Khˆong gian vector
3.1 Kha´i niˆe
.
m vˆe
`
khˆong gian vector
3.1.1 D
-
i
.
nh nghı
˜
a khˆong gian vector
D
-
i
.
nh nghı
˜
a 3.1. Cho mˆo
.
t tˆa
.
p ho
.
.

p E kha´c rˆo
˜
ng va` mˆo
.
t tru
.
`o
.
ng sˆo
´
T cu`ng
v´o
.
i hai phe´p toa´n:
- Phe´p cˆo
.
ng:
E × E −→ E
(x, y) −→ x + y.
- Phe´p nhˆan ngoa`i
T × E −→ E
(λ, x) −→λx.
E cu`ng v´o
.
i hai phe´p toa´n trˆen lˆa
.
p tha`nh mˆo
.
t khˆong gian vector trˆen K, hay
K- khˆong gian vector nˆe

´
u 8 tiˆen d¯ˆe
`
sau d¯ˆay d¯u
.
o
.
.
c thu
.
.
c hiˆe
.
n:
(1) (x + y) + z = x + (y + z); ∀x, y, z ∈ E;
(2) ∃0
E
∈ E sao cho: x + 0
E
= 0
E
+ x = x; ∀x ∈ E;
(3) ∀x ∈ E, ∃ − x ∈ E sao cho: x + (−x) = (−x) + x = 0
E
;
(4) x + y = y + x; ∀x, y ∈ E;
(5) λ(x + y) = λx + λy; ∀x, y ∈ E; ∀λ ∈ K;
(6) (λ + µ)x = λx + µx; ∀x ∈ E; ∀λ, µ ∈ K;
(7) (λµ)x = λ(µx); ∀x ∈ E; ∀λ, µ ∈ K;
47

48 3. Khˆong gian vector
(8) 1x = x, ∀x ∈ K.
Mˆo
˜
i phˆa
`
n tu
.
’
cu
’
a E d¯u
.
o
.
.
c go
.
i la` mˆo
.
t vector, mˆo
˜
i sˆo
´
thuˆo
.
c K go
.
i la` mˆo
.

t vˆo
hu
.
´o
.
ng.
3.1.2 V`ai v´ı du
.
.
a. Tˆa
.
p ho
.
.
p V = Mat
m×n
(K) ca´c ma trˆa
.
n cˆa
´
p m × n trˆen tru
.
`o
.
ng K cu`ng v´o
.
i
phe´p toa´n cˆo
.
ng hai ma trˆa

.
n, nhˆan mˆo
.
t sˆo
´
cu
’
a tru
.
`o
.
ng K v´o
.
i mˆo
.
t ma trˆa
.
n la`
mˆo
.
t K- khˆong gian vector. Vector
−→
0 la` ma trˆa
.
n O, vector d¯ˆo
´
i −A la` ma trˆa
.
n
d¯ˆo

´
i cu
’
a ma trˆa
.
n A.
b. Cho V la` tˆa
.
p ho
.
.
p ca´c vector hı`nh ho
.
c v´o
.
i vector
−→
0 la` vector co´ mod¯un
b˘a
`
ng 0 va` co´ hu
.
´o
.
ng tu`y y´, ta xa´c d¯i
.
nh phe´p cˆo
.
ng va` phe´p nhˆan ngoa`i trˆen V
nhu

.
sau:
Phe´p cˆo
.
ng:
V × V −→ V
(
−→
x ,
−→
y ) −→
−→
x +
−→
y
−→
x +
−→
y d¯u
.
o
.
.
c xa´c d¯i
.
nh theo quy t˘a
´
c hı`nh bı`nh ha`nh
Vector d¯ˆo
´

i −
−→
x la` vector cu`ng phu
.
o
.
ng v´o
.
i vector
−→
x , co´ d¯ˆo
.
da`i b˘a
`
ng d¯ˆo
.
da`i
vector
−→
x va` ngu
.
o
.
.
c hu
.
´o
.
ng v´o
.

i vector
−→
x .
Phe´p nhˆan ngoa`i v´o
.
i mˆo
.
t sˆo
´
: v´o
.
i α ∈ R,
−→
x ∈ V , α
−→
x la` mˆo
.
t vector cu`ng
phu
.
o
.
ng v´o
.
i
−→
x , co´ d¯ˆo
.
da`i b˘a
`

ng tı´ch cu
’
a |α| v´o
.
i d¯ˆo
.
da`i cu
’
a
−→
x va` co´ hu
.
´o
.
ng
cu`ng hu
.
´o
.
ng v´o
.
i
−→
x nˆe
´
u α > 0, ngu
.
o
.
.

c hu
.
´o
.
ng v´o
.
i
−→
x nˆe
´
u α < 0.
Dˆe
˜
thˆa
´
y r˘a
`
ng tˆa
.
p V cu`ng v´o
.
i hai phe´p toa´n trˆen thoa
’
ma
˜
n 8 tiˆen d¯ˆe
`
cu
’
a d¯i

.
nh
nghı
˜
a khˆong gian vector. Vˆa
.
y V la` mˆo
.
t khˆong gian vector trˆen R.
c. Cho tru
.
`o
.
ng K, v´o
.
i n ≥ 1, xe´t tı´ch D
-
ˆeca´c:
K
n
= {(x
1
, x
2
, ..., x
n
)/x
i
∈ K, i = 1, 2, ..., n}
cu`ng hai phe´p toa´n:

(x
1
, x
2
, ..., x
n
) + (y
1
, y
2
, ..., y
n
) = (x
1
+ y
1
, x
2
+ y
2
, ..., x
n
+ y
n
)
k(x
1
, x
2
, ..., x

n
) = (kx
1
, kx
2
, ..., kx
n
), k ∈ K.
Dˆe
˜
thˆa
´
y K
n
cu`ng hai phe´p toa´n trˆen la` mˆo
.
t K− khˆong gian vector. Vector
O = (0, 0, ..., 0), vector d¯ˆo
´
i cu
’
a x = (x
1
, x
2
, ..., x
n
) la` −x = (−x
1
, −x

2
, ..., −x
n
).
D
-
˘a
.
c biˆe
.
t: Khi n = 1 thı` ba
’
n thˆan K cu
˜
ng la` mˆo
.
t K− khˆong gian vector.
d. Tˆa
.
p ho
.
.
p ca´c sˆo
´
thu
.
.
c R v´o
.
i phe´p cˆo

.
ng sˆo
´
thu
.
.
c va` phe´p nhˆan sˆo
´
thu
.
.
c v´o
.
i
sˆo
´
h˜u
.
u ty
’
la` mˆo
.
t Q− khˆong gian vector.
e. Tˆa
.
p K[x] ca´c d¯a th´u
.
c mˆo
.
t biˆe

´
n hˆe
.
sˆo
´
trˆen K v´o
.
i phe´p cˆo
.
ng d¯a th´u
.
c va`
phe´p nhˆan mˆo
.
t phˆa
`
n tu
.
’
thuˆo
.
c tru
.
`o
.
ng K v´o
.
i mˆo
.
t d¯a th´u

.
c la` mˆo
.
t K− khˆong
gian vector.
Ba`i gia
’
ng D
-
a
.
i sˆo
´
tuyˆe
´
n tı´nh
3.1. Kha´ i niˆe
.
m vˆe
`
khˆong gian vector 49
3.1.3 Mˆo
.
t sˆo
´
tı´nh chˆa
´
t d¯o
.
n gia

’
n cu
’
a khˆong gian vector.
Cho V la` mˆo
.
t K− khˆong gian vector tu`y y´. Khi d¯o´, ta luˆon co´:
Tı´nh chˆa
´
t 3.1 (Tı´nh duy nhˆa
´
t cu
’
a phˆa
`
n tu
.
’
khˆong.). Chı
’
co´ duy nhˆa
´
t
mˆo
.
t vector 0 ∈ V sao cho
∀x ∈ V : x + 0 = 0 + x = x.
Thˆa
.
t vˆa

.
y, nˆe
´
u θ cu
˜
ng la` mˆo
.
t vector khˆong cu
’
a V thı`:
θ = θ + 0 = 0.
Tı´nh chˆa
´
t 3.2 (Tı´nh duy nhˆa
´
t cu
’
a phˆa
`
n tu
.
’
d¯ˆo
´
i.). V´o
.
i mˆo
˜
i x ∈ V , tˆo
`

n
ta
.
i duy nhˆa
´
t phˆa
`
n tu
.
’
d¯ˆo
´
i cu
’
a x la` −x sao cho:
x + (−x) = 0.
Thˆa
.
t vˆa
.
y, nˆe
´
u x

cu
˜
ng la` mˆo
.
t vector d¯ˆo
´

i cu
’
a x thı` :
−x = −x + 0 = −x + (x + x

) = (−x + x) + x

= 0 + x

= x

.
Tı´nh chˆa
´
t 3.3. Luˆa
.
t gia
’
n u
.
´o
.
c co´ hiˆe
.
u lu
.
.
c trong V , t´u
.
c la`:

+) (x + z = y + z) ⇒ (x = y), ∀x, y, z ∈ V ;
+) (z + x = z + y) ⇒ (x = y), ∀x, y, z ∈ V.
Thˆa
.
t vˆa
.
y, (x + z = y + z) ⇒ [(x + z) + (−z) = (y + z) + (−z)]
⇒ [x + (z − z) = y + (z − z)] ⇒ (x + 0 = y + 0) ⇒ (x = y).
Tu
.
o
.
ng tu
.
.
cho phˆa
`
n co`n la
.
i.
Tı´nh chˆa
´
t 3.4. ∀x, y, z ∈ V, (x + y = z) ⇔ (x = z − y).
Thˆa
.
t vˆa
.
y, (x + y = z) ⇔ [(x + y) + (−y) = z + (−y)] ⇔ [x + (y − y) = z − y]
⇔ (x + 0 = z − y) ⇔ (x = z − y).
Tı´nh chˆa

´
t 3.5. ∀λ ∈ K, ∀x ∈ V, λx = 0 ⇔

λ = 0 ∈ K
x = 0 ∈ V
Ch´u
.
ng minh. (⇐) λ0 = λ(0 + 0) = λ0 + λ0 ⇒ λ0 = 0 (theo luˆa
.
t gia
’
n u
.
´o
.
c);
0x = (0 + 0)x = 0x + 0x ⇒ 0x = 0 (theo luˆa
.
t gia
’
n u
.
´o
.
c).
(⇒) Gia
’
su
.
’

λx = 0 va` λ = 0. Khi d¯o´ ∃λ
−1
∈ K va` ta co´:
x = 1x = (λ
−1
λ)x = λ
−1
(λx) = λ
−1
0 = 0, t´u
.
c la` x = 0 ∈ V .
Ba`i gia
’
ng D
-
a
.
i sˆo
´
tuyˆe
´
n tı´nh
50 3. Khˆong gian vector
Tı´nh chˆa
´
t 3.6. ∀λ ∈ K, ∀x ∈ V, −(λx) = (−λ)x = λ(−x).
Thˆa
.
t vˆa

.
y,
λx + (−λ)x = [λ + (−λ)]x = 0x = 0 = λx + [−(λx)] ⇒ (−λ)x = −(λx);
λx + λ(−x) = λ[x + (−x)] = λ0 = 0 = λx + [−(λx)] ⇒ λ(−x) = −(λx)
Vˆa
.
y: −(λx) = (−λ)x = λ(−x).
3.2 Khˆong gian vector con.
D
-
i
.
nh nghı
˜
a 3.2. Mˆo
.
t tˆa
.
p ho
.
.
p con W = ∅ cu
’
a K− khˆong gian vector V d¯u
.
o
.
.
c
go

.
i la` khˆong gian vector con cu
’
a V nˆe
´
u W ˆo
’
n d¯i
.
nh d¯ˆo
´
i v´o
.
i phe´p toa´n cˆo
.
ng va`
phe´p nhˆan ngoa`i trˆen V . T´u
.
c la`, x + y ∈ W va` λx ∈ W v´o
.
i mo
.
i x, y ∈ W,
mo
.
i λ ∈ K.
D
-
u
.

o
.
ng nhiˆen khi W la` mˆo
.
t khˆong gian vector con cu
’
a V thı` W cu
˜
ng la`
mˆo
.
t khˆong gian vector trˆen tru
.
`o
.
ng K.
Vı´ du
.
.
(1) K− Khˆong gian vector V la` mˆo
.
t khˆong gian con cu
’
a chı´nh no´ va` d¯u
.
o
.
.
c
go

.
i la` khˆong gian con khˆong thu
.
.
c su
.
.
. Tˆa
.
p ho
.
.
p {0
V
} chı
’
gˆo
`
m mˆo
.
t vector
khˆong cu
˜
ng la` mˆo
.
t khˆong gian vector con cu
’
a V va` d¯u
.
o

.
.
c go
.
i la` khˆong
gian con tˆa
`
m thu
.
`o
.
ng cu
’
a V .
Ta go
.
i khˆong gian con thu
.
.
c su
.
.
cu
’
a V la` mˆo
.
t khˆong gian con kha´c {0
V
}
va` kha´c V .

(2) Nˆe
´
u coi C la` mˆo
.
t R− khˆong gian vector thı` R ⊂ C la` mˆo
.
t khˆong gian
vector con cu
’
a C. Nˆe
´
u coi C la` mˆo
.
t C− khˆong gian vector thı` R khˆong
la` mˆo
.
t khˆong gian vector con cu
’
a C vı` R khˆong ˆo
’
d¯i
.
nh v´o
.
i phe´p nhˆan
v´o
.
i mˆo
.
t sˆo

´
ph´u
.
c.
(3) Tˆa
.
p W = {a
0
+ a
1
x + a
2
x
x
+ · · · + a
n
x
n
|a
i
∈ K} trong d¯o´ n la` mˆo
.
t sˆo
´
tu
.
.
nhiˆen cho tru
.
´o

.
c, la` mˆo
.
t khˆong gian vector con cu
’
a K− khˆong gian vector
K[x].
D
-
i
.
nh ly´ 3.1. Cho W la` mˆo
.
t tˆa
.
p con kha´c rˆo
˜
ng cu
’
a K− khˆong gian vector V .
Khi d¯o´ W la` mˆo
.
t khˆong gian vector con cu
’
a V khi va` chı
’
khi
λx + µy ∈ W, ∀x, y ∈ W, ∀λ, µ ∈ K.
Ch´u
.

ng minh. (⇒) Gia
’
su
.
’
W la` khˆong gian con cu
’
a V .
Khi d¯o´, ∀x, y ∈ W, ∀λ, µ ∈ K do λx, µy ∈ W nˆen λx + µy ∈ W .
(⇐) Cho
.
n λ = µ = 1 thı` ∀x, y ∈ W , ta d¯ˆe
`
u co´ x + y ∈ W ;
Ba`i gia
’
ng D
-
a
.
i sˆo
´
tuyˆe
´
n tı´nh
3.3. Su
.
.
phu
.

thuˆo
.
c tuyˆe
´
n t´ınh v`a d¯ˆo
.
c lˆa
.
p tuyˆe
´
n t´ınh. 51
Cho
.
n λ = 1, µ = 0 thı` ∀x ∈ W, y = x, ta d¯ˆe
`
u co´ λx + 0x = λx ∈ W.
Do d¯o´ W la` mˆo
.
t khˆong gian vector con cu
’
a V .
3.3 Su
.
.
phu
.
thuˆo
.
c tuyˆe
´

n t´ınh v`a d¯ˆo
.
c lˆa
.
p tuyˆe
´
n t´ınh.
3.3.1 Tˆo
’
ho
.
.
p tuyˆe
´
n tı´nh va` biˆe
’
u thi
.
tuyˆe
´
n tı´nh.
D
-
i
.
nh nghı
˜
a 3.3. Cho x
1
, x

2
, ..., x
n
la` n vector (n ≥ 1) cu
’
a K− khˆong gian
vector V va` λ
1
, λ
2
, ..., λ
n
la` n vˆo hu
.
´o
.
ng trong K. Vector
x = λ
1
x
1
+ λ
2
x
2
+ · · · + λ
n
x
n
=

n

i=1
λ
i
x
i
d¯u
.
o
.
.
c go
.
i la` tˆo
’
ho
.
.
p tuyˆe
´
n tı´nh cu
’
a hˆe
.
vector (x
1
, x
2
, ..., x

n
) = (x
i
)
i=1,n
v´o
.
i ho
.
hˆe
.
sˆo
´
(λ
1
, λ
2
, ..., λ
n
) = (λ
i
)
i=1,n
.
Khi vector x la` mˆo
.
t tˆo
’
ho
.

.
p tuyˆe
´
n tı´nh cu
’
a hˆe
.
(x
i
)
i=1,n
thı` ta ba
’
o x biˆe
’
u
thi
.
tuyˆe
´
n tı´nh d¯u
.
o
.
.
c qua hˆe
.
(x
i
)

i=1,n
.
Vı´ du
.
. Cho
−→
x
1
= (1, −2),
−→
x
2
= (3, 1),
−→
x = (5, −3) ∈ R
2
.
Ta co´ 2
−→
x
1
+
−→
x
2
= (5, −3) =
−→
x .
Vˆa
.

y
−→
x la` tˆo
’
ho
.
.
p tuyˆe
´
n tı´nh cu
’
a hˆe
.
(
−→
x
1
,
−→
x
2
), hay
−→
x biˆe
’
u thi
.
tuyˆe
´
n tı´nh d¯u

.
o
.
.
c
qua hˆe
.
(
−→
x
1
,
−→
x
2
).
Nhˆa
.
n xe´t.
(1) Ca´ch biˆe
’
u diˆe
˜
n x =
n

i=1
λ
i
x

i
no´i chung khˆong duy nhˆa
´
t.
Vı´ du
.
. Trong khˆong gian vector thu
.
.
c R
2
, xe´t 3 vector x
1
= (−1, 0), x
2
=
(0, −1), x
3
= (1, 1). Khi d¯o´ vector khˆong 0 = (0, 0) biˆe
’
u thi
.
tuyˆe
´
n tı´nh
d¯u
.
o
.
.

c qua hˆe
.
(x
1
, x
2
, x
3
) b˘a
`
ng ı´t nhˆa
´
t hai ca´ch sau:
0 = 0x
1
+ 0x
2
+ 0x
3
;
0 = 1.x
1
+ 1.x
2
+ 1.x
3
.
(2) Nˆe
´
u x = 0 ∈ V thı` v´o

.
i mo
.
i hˆe
.
vector (x
i
)
i=1,n
⊂ V , x bao gi`o
.
cu
˜
ng biˆe
’
u
thi
.
tuyˆe
´
n tı´nh d¯u
.
o
.
.
c qua (x
i
)
i=1,n
.

Vı´ du
.
. 0 =
n

i=1
λ
i
x
i
, λ
i
= 0, ∀i = 1, n. Trong tru
.
`o
.
ng ho
.
.
p na`y ta no´i
0 biˆe
’
u thi
.
tuyˆe
´
n tı´nh tˆa
`
m thu
.

`o
.
ng qua hˆe
.
trˆen. Nˆe
´
u 0 co´ ı´t nhˆa
´
t hai
ca´ch biˆe
’
u thi
.
tuyˆe
´
n tı´nh qua hˆe
.
(x
i
)
i=1,n
thı` ta no´i 0 biˆe
’
u thi
.
tuyˆe
´
n tı´nh
khˆong tˆa
`

m thu
.
`o
.
ng qua hˆe
.
(x
i
)
i=1,n
.
Ba`i gia
’
ng D
-
a
.
i sˆo
´
tuyˆe
´
n tı´nh
52 3. Khˆong gian vector
3.3.2 D
-
ˆo
.
c lˆa
.
p tuyˆe

´
n t´ınh v`a phu
.
thuˆo
.
c tuyˆe
´
n t´ınh.
D
-
i
.
nh nghı
˜
a 3.4. Hˆe
.
n vector (n ≥ 1) (x
i
)
i=1,n
trong K− khˆong gian vector
V d¯u
.
o
.
.
c go
.
i la` d¯ˆo
.

c lˆa
.
p tuyˆe
´
n tı´nh nˆe
´
u vector khˆong chı
’
co´ duy nhˆa
´
t mˆo
.
t ca´ch
biˆe
’
u thi
.
tuyˆe
´
n tı´nh qua hˆe
.
d¯o´ b˘a
`
ng tˆo
’
ho
.
.
p tuyˆe
´

n tı´nh tˆa
`
m thu
.
`o
.
ng. Hˆe
.
khˆong
d¯ˆo
.
c la
.
p tuyˆe
´
n tı´nh go
.
i la` hˆe
.
phu
.
thuˆo
.
c tuyˆe
´
n tı´nh.
Nhu
.
vˆa
.

y, hˆe
.
(x
i
)
i=1,n
d¯ˆo
.
c lˆa
.
p tuyˆe
´
n tı´nh khi va` chı
’
khi

n

i=1
λ
i
x
i
= 0 ∈ V

⇒ (λ
1
= λ
2
= · · · = λ

n
= 0 ∈ K).
Co`n hˆe
.
(x
i
)
i=1,n
phu
.
thuˆo
.
c tuyˆe
´
n tı´nh nˆe
´
u va` chı
’
nˆe
´
u co´ ı´t nhˆa
´
t mˆo
.
t ho
.
vˆo
hu
.
´o

.
ng (λ
i
)
i=1,n
khˆong d¯ˆo
`
ng th`o
.
i b˘a
`
ng khˆong sao cho
n

i=1
λ
i
x
i
= 0 ∈ V .
Vı´ du
.
.
(1) Cho V = R
3
la` mˆo
.
t R− khˆong gian vector. Xe´t hˆe
.
{x

1
= (1, 1, 1), x
2
= (1, 1, 0), x
3
= (1, 0, 0)}.
Gia
’
su
.
’
tˆo
`
n ta
.
i λ
1
, λ
2
, λ
3
∈ R sao cho:
λ
1
x
1
+ λ
2
x
2

+ λ
3
x
3
= 0 ⇔ (λ
1
+ λ
2
+ λ
3
, λ
1
+ λ
2
, λ
1
) = 0
⇔





λ
1
+ λ
2
+ λ
3
= 0

λ
1
+ λ
2
= 0
λ
1
= 0
⇔





λ
1
= 0
λ
2
= 0
λ
3
= 0
Vˆa
.
y hˆe
.
d¯a
˜
cho d¯ˆo

.
c lˆa
.
p tuyˆe
´
n tı´nh trong R
3
.
(2) Cho V = R
2
la` mˆo
.
t R− khˆong gian vector. Xe´t hˆe
.
3 vector :
{x
1
= (1, −2), x
2
= (1, 4), x
3
= (3, 5)}.
Gia
’
su
.
’
co´ λ
1
, λ

2
, λ
3
∈ R sao cho:
λ
1
x
1
+ λ
2
x
2
+ λ
3
x
3
= 0 ⇔ (λ
1
+ λ
2
+ 3λ
3
, −2λ
1
+ 4λ
2
+ 5λ
3
) = 0
⇔


λ
1
+ λ
2
+ 3λ
3
= 0
−2λ
1
+ 4λ
2
+ 5λ
3
= 0
⇔

λ
1
+ λ
2
= −3λ
3
−2λ
1
+ 4λ
2
= −5λ
3
⇔






λ
1
= −
7
6
λ
3
λ
2
= −
11
6
λ
3
T`u
.
d¯ˆay ta co´ thˆe
’
cho
.
n ra rˆa
´
t nhiˆe
`
u ho

.
vˆo hu
.
´o
.
ng (λ
i
)
i=1,3
khˆong d¯ˆo
`
ng th`o
.
i
b˘a
`
ng khˆong sao cho
3

i=1
λ
i
x
i
= 0
Vˆa
.
y hˆe
.
d¯a

˜
cho phu
.
thuˆo
.
c tuyˆe
´
n tı´nh.
Ba`i gia
’
ng D
-
a
.
i sˆo
´
tuyˆe
´
n tı´nh
3.3. Su
.
.
phu
.
thuˆo
.
c tuyˆe
´
n t´ınh v`a d¯ˆo
.

c lˆa
.
p tuyˆe
´
n t´ınh. 53
Quy u
.
´o
.
c. Hˆe
.
∅ la` hˆe
.
d¯ˆo
.
c lˆa
.
p tuyˆe
´
n tı´nh. Vector 0 ∈ V la` tˆo
’
ho
.
.
p tuyˆe
´
n tı´nh
tˆa
`
m thu

.
`o
.
ng cu
’
a hˆe
.
∅ va` la` vector duy nhˆa
´
t biˆe
’
u thi
.
tuyˆe
´
n tı´nh qua hˆe
.
∅.
Nhˆa
.
n xe´t.
(1) {
−→
0 } la` hˆe
.
phu
.
thuˆo
.
c tuyˆe

´
n tı´nh.
(2) Nˆe
´
u hˆe
.
(
−→
x
i
)
i=1,n
d¯ˆo
.
c lˆa
.
p tuyˆe
´
n tı´nh trong V thı` v´o
.
i mo
.
i
−→
x ∈ V ,
−→
x co´
khˆong qua´ mˆo
.
t ca´ch biˆe

’
u thi
.
tuyˆe
´
n tı´nh qua hˆe
.
(
−→
x
i
)
i=1,n
.
(3) Cho hˆe
.
(
−→
x
i
)
i=1,n
d¯ˆo
.
c lˆa
.
p tuyˆe
´
n tı´nh trong V va`
−→

x ∈ V , nˆe
´
u
−→
x biˆe
’
u thi
.
tuyˆe
´
n tı´nh d¯u
.
o
.
.
c qua hˆe
.
(
−→
x
i
)
i=1,n
thı` ca´ch biˆe
’
u diˆe
˜
n d¯o´ la` duy nhˆa
´
t.

Ch´u
.
ng minh. Gia
’
su
.
’
−→
x biˆe
’
u thi
.
tuyˆe
´
n tı´nh d¯u
.
o
.
.
c qua hˆe
.
(
−→
x
i
)
i=1,n
t´u
.
c la`

tˆo
`
n ta
.
i ca´c λ
i
∈ K sao cho
−→
x = λ
1
−→
x
1
+ λ
2
−→
x
2
+ · · · + λ
n
−→
x
n
.
Nˆe
´
u ngoa`i ca´c λ
i
trˆen co`n tˆo
`

n ta
.
i ca´c µ
i
∈ K sao cho
−→
x = µ
1
−→
x
1
+ µ
2
−→
x
2
+ · · · + µ
n
−→
x
n
.
Thı` ta co´:
λ
1
−→
x
1
+ λ
2

−→
x
2
+ · · · + λ
n
−→
x
n
= µ
1
−→
x
1
+ µ
2
−→
x
2
+ · · · + µ
n
−→
x
n
⇔ (λ
1
− µ
1
)
−→
x

1
+ (λ
2
− µ
2
)
−→
x
2
+ · · · + (λ
n
− µ
n
)x
n
=
−→
0
⇒












λ
1
− µ
1
= 0
λ
2
− µ
2
= 0
· · ·
λ
n
− µ
n
= 0
(do hˆe
.
(
−→
x
i
)
i=1,n
d¯ˆo
.
c lˆa
.
p tuyˆe
´

n tı´nh)
⇔ λ
i
= µ
i
, ∀i = 1, n. Vˆa
.
y su
.
.
biˆe
’
u thi
.
tuyˆe
´
n tı´nh cu
’
a
−→
x qua hˆe
.
(
−→
x
i
)
i=1,n
la` duy nhˆa
´

t.
3.3.3 V`ai t´ınh chˆa
´
t vˆe
`
hˆe
.
phu
.
thuˆo
.
c tuyˆe
´
n t´ınh v`a hˆe
.
d¯ˆo
.
c lˆa
.
p tuyˆe
´
n
t´ınh.
Tı´nh chˆa
´
t 3.7. (i) Hˆe
.
gˆo
`
m mˆo

.
t vector {
−→
x } d¯ˆo
.
c lˆa
.
p tuyˆe
´
n tı´nh khi va` chı
’
khi
−→
x =
−→
0 .
(ii) Mo
.
i hˆe
.
vector ch´u
.
a
−→
0 d¯ˆe
`
u phu
.
thuˆo
.

c tuyˆe
´
n tı´nh.
Tı´nh chˆa
´
t na`y kha´ d¯o
.
n gia
’
n, ba
.
n d¯o
.
c tu
.
.
ch´u
.
ng minh.
Ba`i gia
’
ng D
-
a
.
i sˆo
´
tuyˆe
´
n tı´nh