3 Khˆ ong gian vector
3.1.2 V`ai v´ı du
a. Tˆa.p ho..p V = Matm×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 ) 7−→−→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:
Kn ={(x1, x2, ..., xn)/xi ∈ K, i= 1,2, ..., n}
cu`ng hai phe´ p toa´ n:
(x1, x2, ..., xn) + (y1, y2, ..., yn) = (x1+y1, x2+y2, ..., xn +yn)
k(x1, x2, ..., xn) = (kx1, kx2, ..., kxn), k ∈ K.
Dˆe˜ thˆa´y Kn 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’ax = (x1, x2, ..., xn) la`−x = (−x1,−x2, ...,−xn).
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.
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 phˆ` n tu.a ’ 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 phˆ` n tu.a ’ d¯ˆo´i.). V´o.i mˆo˜i x ∈ V, tˆ` no 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 x0 cu˜ ng la` mˆo.t vector d¯ˆo´i cu’a x thı` :
−x = −x + 0 =−x+ (x+x0) = (−x+x) +x0 = 0 +x0 = x0.
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 6= 0. Khi d¯o´ ∃λ−1 ∈ K va` ta co´ :
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 6=∅ cu’a K− khˆong gian vector V d¯u.o.. c go.i la` khˆong gian vector con cu’a Vnˆe´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 {0V} 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 tˆ` m thu.`o.nga cu’a V.
Ta go.i khˆong gian con thu..c su.. cu’a V la` mˆo.t khˆong gian con kha´c {0V}
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 = {a0+a1x+a2xx+· · ·+anxn|ai ∈ 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;
3.3. Su.. phu. thuˆo.c tuyˆe´n t´ınh v`a d¯ˆo.c lˆa.p tuyˆe´n t´ınh. 51Cho.n λ = 1, µ = 0 thı` ∀x ∈ W, y = x, ta d¯ˆe` u co´ λx+ 0x =λx ∈ W. 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 x1, x2, ..., xn la` n vector (n ≥ 1) cu’a K− khˆong gian vector V va` λ1, λ2, ..., λn la` n vˆo hu.´o.ng trong K. Vector
x =λ1x1+λ2x2+· · ·+λnxn =
n
X
i=1
λixi
d¯u.o.. c go.i la` tˆo’ ho.. p tuyˆe´n tı´nh cu’a hˆe. vector (x1, x2, ..., xn) = (xi)i=1,n v´o.i ho. hˆe. sˆo´ (λ1, λ2, ..., λn) = (λi)i=1,n.
Khi vector x la` mˆo.t tˆo’ ho..p tuyˆe´n tı´nh cu’a hˆe. (xi)i=1,n thı` ta ba’o x biˆe’u thi. tuyˆe´n tı´nh d¯u.o.. c qua hˆe. (xi)i=1,n.
Vı´ du.. Cho −→x1 = (1,−2), −→x2 = (3,1),−→x = (5,−3) ∈ R2. Ta co´ 2−→x1 +−→x2 = (5,−3) = −→x.
Vˆa.y −→x la` tˆo’ ho.. p tuyˆe´n tı´nh cu’a hˆe. (−→x1,−→x2), hay −→x biˆe’u thi. tuyˆe´n tı´nh d¯u.o..c qua hˆe. (−→x1,−→x2). Nhˆa.n xe´ t. (1) Ca´ ch biˆe’u diˆe˜n x = n P i=1
λixi no´ i chung khˆong duy nhˆa´t.
Vı´ du.. Trong khˆong gian vector thu.. c R2, xe´ t 3 vector x1 = (−1,0), x2 = (0,−1), x3 = (1,1). Khi d¯o´ vector khˆong 0 = (0,0) biˆe’u thi. tuyˆe´n tı´nh d¯u.o.. c qua hˆe. (x1, x2, x3) b˘a`ng ı´t nhˆa´t hai ca´ ch sau:
0 = 0x1+ 0x2 + 0x3; 0 = 1.x1 + 1.x2+ 1.x3.
(2) Nˆe´u x = 0 ∈ V thı` v´o.i mo.i hˆe. vector (xi)i=1,n ⊂ V, x bao gi`o. cu˜ ng biˆe’u thi. tuyˆe´n tı´nh d¯u.o..c qua (xi)i=1,n.
Vı´ du.. 0 =
n
P
i=1
λixi, λi = 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. (xi)i=1,n thı` ta no´ i 0 biˆe’u thi. tuyˆe´n tı´nh khˆong tˆa` m thu.`o.ng qua hˆe. (xi)i=1,n.
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) (xi)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. (xi)i=1,n d¯ˆo.c lˆa.p tuyˆe´n tı´nh khi va` chı’ khi Xn
i=1
λixi = 0 ∈ V⇒ (λ1 =λ2 =· · · =λn = 0 ∈ K).
Co`n hˆe. (xi)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 (λi)i=1,n khˆong d¯ˆo` ng th`o.i b˘a`ng khˆong sao cho Pn
i=1
λixi = 0 ∈ V. Vı´ du..
(1) Cho V =R3 la` mˆo.t R− khˆong gian vector. Xe´ t hˆe.
{x1 = (1,1,1), x2 = (1,1,0), x3 = (1,0,0)}.
Gia’ su.’ tˆo` n ta.i λ1, λ2, λ3 ∈ R sao cho:
λ1x1+λ2x2 +λ3x3 = 0 ⇔(λ1+λ2+λ3, λ1+λ2, λ1) = 0 ⇔ λ1+λ2+λ3 = 0 λ1+λ2 = 0 λ1 = 0 ⇔ λ1 = 0 λ2 = 0 λ3 = 0 Vˆa.y hˆe. d¯a˜ cho d¯ˆo.c lˆa.p tuyˆe´n tı´nh trong R3.
(2) Cho V =R2 la` mˆo.t R− khˆong gian vector. Xe´ t hˆe. 3 vector :
{x1 = (1,−2), x2 = (1,4), x3 = (3,5)}.
Gia’ su.’ co´ λ1, λ2, λ3 ∈ R sao cho:
λ1x1+λ2x2 +λ3x3 = 0 ⇔(λ1+λ2+ 3λ3,−2λ1 + 4λ2+ 5λ3) = 0 ⇔ ( λ1+λ2+ 3λ3 = 0 −2λ1+ 4λ2+ 5λ3 = 0 ⇔ ( λ1+λ2 = −3λ3 −2λ1+ 4λ2 = −5λ3 ⇔ λ1 =−7 6λ3 λ2 =−11 6 λ3 T`u. d¯ˆay ta co´ thˆe’ cho.n ra rˆa´t nhiˆe` u ho. vˆo hu.´o.ng (λi)i=1,3 khˆong d¯ˆo` ng th`o.i b˘a`ng khˆong sao cho P3
i=1
λixi = 0
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. (−→xi)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. (−→xi)i=1,n.
(3) Cho hˆe. (−→xi)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. (→−xi)i=1,n t´u.c la` tˆo` n ta.i ca´c λi ∈ K sao cho
−
→x =λ1−→x1 +λ2x→−2 +· · ·+λn−→xn.
Nˆe´u ngoa`i ca´ c λi trˆen co`n tˆo` n ta.i ca´c µi ∈ K sao cho
− →x = µ1−→x1 +µ2→−x2 +· · ·+µn−→xn. Thı` ta co´ : λ1−→x1 +λ2−→x2 +· · ·+λ n−→x n =µ1−→x1 +µ2−→x2 +· · ·+µ n−x→ n ⇔(λ1−µ1)x→−1 + (λ2−µ2)−→x2+· · ·+ (λn −µn)xn =−→0 ⇒ λ1−µ1 = 0 λ2−µ2 = 0 · · · λn −µn = 0
(do hˆe. (−→xi)i=1,n d¯ˆo.c lˆa.p tuyˆe´n tı´nh)
⇔ λi = µi, ∀i = 1, n. Vˆa.y su.. biˆe’u thi. tuyˆe´n tı´nh cu’a −→xqua hˆe. (−→xi)i=1,n la` duy nhˆa´t.
3.3.3 V`ai t´ınh chˆa´t vˆ` hˆe. phu. thuˆo.c tuyˆe´n t´ınh v`a hˆe. d¯ˆo.c lˆa.p tuyˆe´ne 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 6=−→0 .
(ii) Mo.i hˆe. vector ch´u.a −→0 d¯ˆe` u phu. thuˆo.c tuyˆe´n tı´nh.
Tı´nh chˆa´t 3.8. V´o.i hˆe. vector (xi)i∈I tuy` y´ (I la` mˆo.t tˆa.p ho.. p bˆa´t ky` kha´ c rˆo˜ng), hˆe. (xi)i∈J go.i la` hˆe. con cu’a hˆe. (xi)i∈I nˆe´u J ⊂ I. Khi d¯o´ :
(i) Nˆe´u hˆe. (xi)i=1,n d¯ˆo.c lˆa.p tuyˆe´n tı´nh thı` mo.i hˆe. con cu’a no´ cu˜ng d¯ˆo.c lˆa.p tuyˆe´n tı´nh.
(ii) Nˆe´u co´ ı´t nhˆa´t mˆo.t hˆe. con phu. thuˆo.c tuyˆe´n tı´nh thı` hˆe. (xi)i=1,n cu˜ ng phu. thuˆo.c tuyˆe´n tı´nh.
Ch´u.ng minh. Gia’ su.’ (xi)i=1,n la` hˆe. d¯ˆo.c lˆa.p tuyˆe´n tı´nh va` (xj)j∈J la` mˆo.t hˆe. con tuy` y´ cu’a no´ , t´u.c la` J ⊂ I = {1,2, ..., n}. Ta cˆa` n ch´u.ng to’ (xj)j∈J d¯ˆo.c lˆa.p tuyˆe´n tı´nh.
Thˆa.t vˆa.y, nˆe´u P
j∈J
λjxj = 0 la` mˆo.t tˆo’ ho..p tuyˆe´n tı´nh b˘a`ng 0 cu’a hˆe. (xj)j∈J
thı` 0 = P
j∈J
λjxj + P
i∈I\J
0.xi la` mˆo.t tˆo’ ho..p tuyˆe´n tı´nh b˘a`ng 0 cu’a hˆe. (xi)i=1,n. Ma` hˆe. (xi)i=1,n la` hˆe. d¯ˆo.c lˆa.p tuyˆe´n tı´nh, suy ra λj = 0, ∀j ∈ J, t´u.c la` (xj)j∈J
d¯ˆo.c lˆa.p tuyˆe´n.
Vı` kha´ i niˆe.m hˆe. phu. thuˆo.c tuyˆe´n tı´nh la` phu’ d¯i.nh cu’a kha´i niˆe.m hˆe. d¯ˆo.c lˆa.p tuyˆe´n tı´nh nˆen hai kh˘a’ng d¯i.nh trong tı´nh chˆa´t na`y la` tu.o.ng d¯u.o.ng nhau. D- i.nh ly´ 3.2 (D- i.nh ly´ d¯˘a.c tru.ng cu’a hˆe. phu. thuˆo.c tuyˆe´n tı´nh). Hˆe. n
vector (n ≥ 2) (xi)i=1,n phu. thuˆo.c tuyˆe´n tı´nh khi va` chı’ khi co´ (ı´t nhˆa´t) mˆo.t vector cu’a hˆe. biˆe’u thi. tuyˆe´n tı´nh d¯u.o.. c qua ca´ c vector co`n la.i.
Ch´u.ng minh. (⇒) Gia’ su.’ hˆe. (xi)i=1,n phu. thuˆo.c tuyˆe´n tı´nh. Lu´c d¯o´ co´ ı´t nhˆa´t mˆo.t ho. vˆo hu.´o.ng (λi)i=1,n khˆong d¯ˆo` ng th`o.i triˆe.t tiˆeu sao cho 0 = Pn
i=1
λixi. Gia’ su.’ λj 6= 0 ∈ K (1≤ j ≤ n). Khi d¯o´
n X i=1 λixi ⇒ −λjxj = X i6=j λixi ⇒ xj =X i6=j −λi λj xi;
t´u.c la` xj biˆe’u thi. tuyˆe´n tı´nh d¯u.o..c qua hˆe. ca´c vector co`n la.i (xi)i∈{1,2,...,n}\{j}. (⇐) Ngu.o.. c la.i, gia’ su’ co. ´ mˆo.t vector cu’a hˆe. ch˘a’ng ha.n xj (1 ≤ j ≤ n), biˆe’u thi. tuyˆe´n tı´nh d¯u.o..c qua hˆe. ca´c vector co`n la.i, t´u.c la` co´ ca´c vˆo hu.´o.ng
λi, i∈ {1,2, ..., n}\{j} sao cho xj = P i6=j λixi. Khi d¯o´ xj =X i6=j λixi ⇒0 = X i6=j λixi + (−1)xj
D- ˆay la` mˆo.t tˆo’ ho..p tuyˆe´n tı´nh khˆong tˆa`m thu.`o.ng b˘a`ng 0 cu’a hˆe. (xi)i=1,n. Vˆa.y hˆe. (xi)i=1,n phu. thuˆo.c tuyˆe´n tı´nh.
3.4. Ha.ng cu˙’a mˆo.t hˆe. vector. 55
3.4 Ha.ng cu˙’a mˆo.t hˆe. vector.
3.4.1 Hˆe. con d¯ˆo.c lˆa.p tuyˆe´n t´ınh tˆo´i d¯a.i.
D- i.nh nghı˜a 3.5. Gia’ su.’ I la` mˆo.t tˆa.p ho..p h˜u.u ha.n va` J ⊂ I. Cho hˆe. vector (xi)i∈I tu`y y´ trong mˆo.t K− khˆong gian vector na`o d¯o´ . Hˆe. (xj)j∈J go.i la` hˆe. con d¯ˆo.c lˆa.p tuyˆe´n tı´nh tˆo´i d¯a.i cu’a hˆe. d¯a˜ cho nˆe´u no´ d¯ˆo.c lˆa.p tuyˆe´n tı´nh va` nˆe´u thˆem bˆa´t ky` vector xi na`o, i ∈ I\J, va`o hˆe. con d¯o´ ta d¯ˆe` u nhˆa.n d¯u.o..c mˆo.t hˆe. phu. thuˆo.c tuyˆe´n tı´nh.
Vı´ du.. Trong R3 cho hˆe. 3 vector {x1 = (1,2,3), x2 = (2,4,6), x3 = (3,6,9)}. Khi d¯o´ mˆo˜i hˆe. 1 vector {x1},{x2},{x3} d¯ˆe` u la` hˆe. con d¯ˆo.c lˆa.p tuyˆe´n tı´nh cu’a hˆe. d¯a˜ cho. Ho.n n˜u.a, x3 = 3x1, x2 = 2x1, x3 = 3
2x2 nˆen ca´ c hˆe. con
{x1, x2},{x1, x3},{x2, x3} d¯ˆe` u phu. thuˆo.c tuyˆe´n tı´nh. Vˆa.y {x1},{x2},{x3} la` ca´ c hˆe. con d¯ˆo.c lˆa.p tuyˆe´n tı´nh tˆo´i d¯a.i cu’a hˆe. {x1, x2, x3} d¯a˜ cho.
Tı´nh chˆa´t 3.9. Nˆe´u hˆe. con (xi)i=1,n cu’a hˆe. (xi)i∈I ({1,2, ..., n} ⊂ I) la` mˆo.t hˆe. con d¯ˆo.c lˆa.p tuyˆe´n tı´nh tˆo´i d¯a.i thı` mo.i vector xi, i ∈ I d¯ˆe` u biˆe’u thi. tuyˆe´n tı´nh d¯u.o.. c qua hˆe. con d¯o´ va` ca´ ch biˆe’u thi. la` duy nhˆa´t.
Tı´nh chˆa´t na`y la` hˆe. qua’ tru..c tiˆe´p cu’a D- i.nh nghı˜a 3.5 va` D-i.nh ly´ 3.2. Bˆo’ d¯ˆ` 3.1 (Bˆe o’ d¯ˆ` co. ba’n vˆee ` su.. phu. thuˆo.c tuyˆe´n tı´nh). Cho
(x1, x2, ..., xm) va` (y1, y2, ..., yn) la` hai hˆe. vector trong khˆong gian vector V. Gia’ su.’ hˆe. (xi)i=1,m d¯ˆo.c lˆa.p tuyˆe´n tı´nh va` mˆo˜i xi (i = 1, m) d¯ˆe` u biˆe’u thi. tuyˆe´n tı´nh d¯u.o.. c qua hˆe. (yj)j=1,n. Khi d¯o´ m ≤n.
D- i.nh ly´ 3.3. Mo.i hˆe. con d¯ˆo.c lˆa.p tuyˆe´n tı´nh tˆo´i d¯a.i cu’a mˆo.t hˆe. h˜u.u ha.n vector trong mˆo.t K− khˆong gian vector tu`y y´ d¯ˆe` u co´ sˆo´ vector b˘a`ng nhau.
Ch´u.ng minh. Gia’ su.’ (xi)i∈I la` mˆo.t hˆe. vector h˜u.u ha.n. Nˆe´u xi = 0 v´o.i mo.i
i ∈ I thı` (xi)i∈I chı’ co´ mˆo.t hˆe. d¯ˆo.c lˆa.p tuyˆe´n tı´nh tˆo´i d¯a.i duy nhˆa´t la` ∅ va` kh˘a’ng d¯i.nh cu’a d¯i.nh ly´ la` hiˆe’n nhiˆen.
Gia’ su.’ hˆe. (xi)i∈I co´ ch´u.a vector kha´ c khˆong. Khi d¯o´ ca´ c hˆe. con d¯ˆo.c lˆa.p tuyˆe´n tı´nh tˆo´i d¯a.i cu’a (xi)i∈I co´ ı´t nhˆa´t mˆo.t vector. Gia’ su.’ (xj)j∈J1 va` (xj)j∈J2
la` hai hˆe. con d¯ˆo.c lˆa.p tuyˆe´n tı´nh tˆo´i d¯a.i cu’a (xi)i∈I (J1 ⊂ I, J2 ⊂ I) v´o.i sˆo´ vector lˆa` n lu.o..t la` m va` n (m, n ≥ 1). Vı` (xj)j∈J2 d¯ˆo.c lˆa.p tuyˆe´n tı´nh tˆo´i d¯a.i nˆen mo.i xj, j ∈ J1 d¯ˆe` u biˆe’u thi. tuyˆe´n tı´nh d¯u.o..c qua (xj)j∈J2. Ma` (xj)j∈J1
d¯ˆo.c lˆa.p tuyˆe´n tı´nh, do d¯o´ theo Bˆo’ d¯ˆe` 3.1, ta co´ m ≤ n. Tu.o.ng tu.. cu˜ ng co´
3.4.2 Ha.ng cu˙’a mˆo.t hˆe. vector.
D- i.nh nghı˜a 3.6. Cho V la` mˆo.t K− khˆong gian vector, (xi)i∈I la` mˆo.t hˆe. vector bˆa´t ky` trong V. Nˆe´u hˆe. con d¯ˆo.c lˆa.p tuyˆe´n tı´nh tˆo´i d¯a.i cu’a (xi)i∈I co´ sˆo´ phˆa` n tu.’ h˜u.u ha.n b˘a`ng r thı` r d¯u.o.. c go.i la` ha.ng cu’a hˆe. (xi)i∈I.
Kı´ hiˆe.u: rank((xi)i∈I) = r.
Vı´ du.. Xe´ t la.i hˆe. vector {x1 = (1,2,3), x2 = (2,4,6), x3 = (3,6,9)} cu’a
R3. Vı` {x1} la` mˆo.t hˆe. con d¯ˆo.c lˆa.p tuyˆe´n tı´nh tˆo´i d¯a.i cu’a hˆe. {x1, x2, x3} nˆen