Không gian Euclide Rn

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) d
u
.
o
.
.
cgo
.
il`akhˆong gian thu
.
.
c (ph´u
.
c) n-chiˆe
`

u v`a d
u
.
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 d
u
.
o
.
.
cgo
.
il`ad
iˆe
’
m hay vecto
.
cu
’
a R

n
(C
n
). C´ac sˆo
´
x
1
,...,x
n
du
.
o
.
.
c
go
.
il`ato
.
ad
ˆo
.
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 d
u
.
o
.
.
cx´et l`a khˆong
gian ph´u
.
c.
Theo d
i
.
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
.
d
ˆo
´
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 d
i
.
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
d
i
.
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-VIII 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) d
u
.
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 d
u
.
o
.
.
c qua c´ac vecto
.
(5.3).
D
-
i
.
nh ngh˜ıa 5.1.2. 1
+
Hˆe
.
vecto
.

(5.3) d
u
.
o
.
.
cgo
.
il`ahˆe
.
d
ˆo
.
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) d
u
.
o
.
.
c
tho
’
a m˜an.
Sˆo
´
nguyˆen du
.
o
.
ng r d
u
.
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) d
ˆe
`
u phu
.
thuˆo
.
c
tuyˆe
´
n t´ınh.
D

ˆe
’
t`ım ha
.
ng cu
’
ahˆe
.
vecto
.
ta lˆa
.
p ma trˆa
.
n c´ac to
.
ad
ˆo
.
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) d
ltt hay pttt ta cˆa
`
nlˆa
.
p ma trˆa
.
n
to
.
ad
ˆo
.

A cu
’
ach´ung v`a t´ınh r(A):
1) Nˆe
´
u r(A)=m th`ı hˆe
.
(5.3) d
ˆo
.
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
˜
nd
u
.
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. D
ˆe
’
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 d
ˆo
.
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 d
ˆo
.
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
’
ad
i
.
nh th´u
.
c con
∆=





14
23






= −5 =0
n˘a
`
mo
.
’
hai h`ang d
ˆa
`
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
.
d
ˆe
`
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 d
i
.
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).
(D
S. ∀ 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 d
ˆay 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.
D
ˆo
´
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
.
.
cd
a
.
i c´ac vecto
.
d
ltt trong c´ac hˆe
.
vecto
.
sau d
ˆay
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
.
ad
ˆo
.
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 d
ˆay 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).
(D
S. 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 d
ˆay 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).
(D
S. 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
.
d
ˆo
.
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
.
ad
ˆo
.
cu
’
ach´ung.
D
-
i
.
nh l´y 5.2.1. Trong khˆong gian R
n
:
1) To
.
ad
ˆo
.
cu
’
amˆo
.
t vecto
.

d
ˆo
´
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
.
d
ˆe
`
ulˆa
.
p th`anh co
.
so
.
’
cu
’
a khˆong gian
R
n
.
Ta x´et vˆa
´
nd
ˆe
`
: Khi co

.
so
.
’
thay d
ˆo
’
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
.
ad
ˆo
.
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
.
yd
ˆe
’
t`ım ma trˆa
.
n chuyˆe
’
nt`u
.
co
.
so
.
’
c˜u sang co
.

so
.
’
m´o
.
id
ˆa
`
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
.
ad
ˆo
.
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
.
ad
ˆo
.
cu
’
ac`ung mˆo
.
t vecto
.
d
ˆo
´
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 du
.
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
d
i
.
nh l´y 1 (phˆa
`
n 2) c´ac vecto
.
n`ay lˆa
.
p th`anh co
.
so
.
’
cu
’
a R
3
.
2) D
ˆe
’
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
.
ad
ˆo
.
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
.
192 Chu
.
o
.

ng 5. Khˆong gian Euclide
R
n
Gia
’
i. Ta c´o thˆe
’
t`ım ha
.
ng cu
’
ahˆe
.
ba vecto
.
d
˜a cho. Ta c´o



10−2
4 −15
13 4



−→




10−2
0 −113
03 6



−→



10−2
0 −113
00 45



.
T`u
.
d
´o suy ra r˘a
`
ng ha
.
ng cu
’
ahˆe
.
vecto
.

d
˜a cho b˘a
`
ng 3 v`a do vˆa
.
yhˆe
.
d´o
l`a d
ˆo
.
clˆa
.
p tuyˆe
´
n t´ınh. Theo di
.
nh l´y 1 n´o lˆa
.
p th`anh mˆo
.
tco
.
so
.
’
. 
V´ı d u
.
4. Gia

’
su
.
’
trong co
.
so
.
’
E
1
, E
2
vecto
.
x c´o to
.
ad
ˆo
.
l`a 1;−2. T`ım
to
.
ad
ˆo
.
cu
’
a vecto
.

d
´o trong co
.
so
.
’
E
1
= E
1
, E
2
= E
1
+ E
2
.
Gia
’
i. D
ˆa
`
u tiˆen ta viˆe
´
t ma trˆa
.
n chuyˆe
’
nt`u
.

co
.
so
.
’
E
1
, E
2
dˆe
´
n E
1
,E
2
.
Ta c´o
E
1
=1· e
1
+0· e
2
,
E
2
=1· e
1
+1· e
2

.
Do d
´o
T =

11
021

⇒ T
−1
=

1 −1
01

.
´
Ap du
.
ng cˆong th´u
.
c (11*) ta c´o

y
1
y
2

= T
−1


x
1
x
2

=

1 −1
01

1
−2

=

3
−2

.
Do d
´o y
1
=3,y
2
= −2. 
V´ı d u
.
5 (ph´ep quay tru
.

cto
.
ad
ˆo
.
). H˜ay dˆa
˜
n ra cˆong th´u
.
cbiˆe
´
nd
ˆo
’
i c´ac
to
.
ad
ˆo
.
cu
’
a vecto
.
trong R
2
trong mˆo
.
tco
.

so
.
’
thu d
u
.
o
.
.
ct`u
.
co
.
so
.
’
ch´ınh
t˘a
´
c e
1
=(1, 0), e
2
=(0, 1) sau ph´ep quay tru
.
cto
.
adˆo
.
g´oc ϕ.

5.2. Co
.
so
.
’
.D
-
ˆo
’
ico
.
so
.
’
193
H`ınh 5.1
Gia
’
i. T`u
.
h`ınh v˜e suy ra r˘a
`
ng vecto
.
e
∗
1
lˆa
.
pv´o

.
i c´ac vecto
.
e
1
v`a e
2
c´ac g´oc tu
.
o
.
ng ´u
.
ng b˘a
`
ng ϕ v`a ϕ−
π
2
.Dod
´oto
.
adˆo
.
cu
’
a e
∗
1
trong co
.

so
.
’
e
1
,e
2
l`a cos ϕ v`a cos

ϕ −
π
2

= sin ϕ:
e
∗
1
= cos ϕ · e
1
+ sin ϕ · e
2
Vecto
.
e
∗
2
lˆa
.
pv´o
.

i e
1
v`a e
2
c´ac g´oc tu
.
o
.
ng ´u
.
ng b˘a
`
ng
π
2
+ ϕ v`a ϕ.Dod
´o
to
.
ad
ˆo
.
cu
’
a n´o trong co
.
so
.
’
e

1
,e
2
l`a cos

π
2
+ ϕ

= − sin ϕ v`a cos ϕ:
e
∗
2
= − sin ϕ · e
1
+ cos ϕ · e
2
.
Nhu
.
vˆa
.
y
e
∗
1
= cos ϕ · e
1
+ sin ϕ · e
2

,
e
∗
2
= − sin ϕ · e
1
+ cos ϕ · e
2
.
v`a t`u
.
d
´o
T
ee
∗
=

cos ϕ − sin ϕ
sin ϕ cos ϕ

T
−1
ee
∗
=

cos ϕ sin ϕ
− sin ϕ cos ϕ


.
Do vˆa
.
y c´ac to
.
ad
ˆo
.
cu
’
a vecto
.
trong co
.
so
.
’
c˜u v`a m´o
.
i liˆen hˆe
.
bo
.
’
i c´ac hˆe
.
th´u
.
c
x = x

∗
cos ϕ − y
∗
sin ϕ,
y = x
∗
sin ϕ + y
∗
cos ϕ.

x
∗
= x cos ϕ + y sin ϕ,
y
∗
= −x sin ϕ + y cos ϕ.


194 Chu
.
o
.
ng 5. Khˆong gian Euclide
R
n
V´ı d u
.
6. Gia
’
su

.
’
x =(3,−1, 0) l`a vecto
.
cu
’
a R
3
v´o
.
ico
.
so
.
’
E
1
, E
2
, E
3
.
T`ım to
.
ad
ˆo
.
cu
’
a x dˆo

´
iv´o
.
ico
.
so
.
’
E
1
=2E
1
−E
2
+3E
3
,
E
2
= E
1
+ E
3
,
E
3
= −E
2
+2E
3

.
Gia
’
i. T`u
.
c´ac khai triˆe
’
n E
1
,E
2
v`a E
3
theo co
.
so
.
’
E
1
,E
2
,E
3
ta c´o ma
trˆa
.
n chuyˆe
’
n

T =



210
−10−1
312



t`u
.
co
.
so
.
’
E
1
,E
2
,E
3
sang co
.
so
.
’
E
1

,E
2
,E
3
.
Ta k´y hiˆe
.
u x
1
,x
2
,x
3
l`a to
.
adˆo
.
cu
’
a x trong co
.
so
.
’
E
1
,E
2
,E
3

.Tac´o



x
1
x
2
x
3



= T
−1



3
−1
0



V`ı T
−1
=




1 −2 −1
−14 2
−11 1



nˆen



x
1
x
2
x
3



=



1 −2 −1
−14 2
−11 1







3
−1
0



=



5
−7
−4



.
Vˆa
.
y trong co
.
so
.
’
m´o
.
i E
1

,E
2
,E
3
ta c´o
x =(5,−7,−4). 
V´ı d u
.
7. Trong khˆong gian R
2
cho co
.
so
.
’
E
1
,E
2
v`a c´ac vecto
.
E
1
=
e
1
− 2e
2
, E
2

=2e
1
+ e
2
, x =3e
1
− 4e
2
.
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
.
ad
ˆo
.
l`a (3,−4). Gia
’
su
.

’
trong
co
.
so
.
’
E
1
,E
2
vecto
.
x c´o to
.
ad
ˆo
.
(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+ D
ˆe
’
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
·

(II) 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 d
u
.
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 d
ˆay 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
.
d
o
.
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 d
u
.
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 d
ˆa 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 d
ˆe
’
chı
’
c´ac vecto
.
d
o
.
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
.
(D
S. 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
.
(D
S.



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

.
(D
S.





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
.
(D
S. 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
.
(D
S. 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
.
ad
ˆo
.
l`a (4, 0,−12). T`ım to
.
a
d
ˆo
.
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
.
ad
ˆo
.
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 d
u
.
o
.
.
cgo
.
i l`a khˆong gian Euclid nˆe
´
u trong
V d
u
.
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
d
ˆa y
(I) x, y = y, x;
(II) x + y, z = x, z + y, z;
(III) 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
),