3 Khˆ ong gian vector
3.3 Su phu thuˆo.c tuyˆe´n t´ınh v`a d¯ˆo.c lˆa.p tuyˆe´n t´ınh
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 rank(x1, x2, x3) = 1.
Nhˆa.n xe´ t. Khi cho (s) = (xi)i=1,n la` mˆo.t hˆe. vector trong V va` r =rank(s) thı`:
(i) r ≤ n,
(ii) Nˆe´u (s) = (xi)i=1,n d¯ˆo.c lˆa.p tuyˆe´n tı´nh thı` rank(s) = r = n. 3.4.3 C´ac hˆe. vector trong Kn.
Trong khˆong gian Kn xe´ t m vector sau:
a1 = (a11, a12, ..., a1n)
a2 = (a21, a22, ..., a2n)
...
am = (am1, am2, ..., amn)
Go.i A = (aij)m×n la` ma trˆa.n cˆa´p m × n trˆen K ma` ca´ c do`ng chı´nh la`
a1, a2, ..., am. Khi d¯o´ ta co´ ca´ c kh˘a’ng d¯i.nh sau d¯ˆay:
D- i.nh ly´ 3.4. V´o.i hˆe. (a1, a2, ..., am) va` ma trˆa.n A d¯u.o.. c d¯i.nh nghı˜a nhu. trˆen, ta co´ :
(1) Hˆe. (a1, a2, ..., am) d¯ˆo.c lˆa.p tuyˆe´n tı´nh trong Kn ⇔ rank(A) =m.
(2) Hˆe. (a1, a2, ..., am) phu. thuˆo.c tuyˆe´n tı´nh trong Kn ⇔ rank(A) = m.
D- i.nh ly´ 3.5. Ha.ng cu’a mˆo.t ma trˆa.n cˆa´p m × n trˆen K b˘a`ng ha.ng cu’a hˆe. vector cˆo.t (tu.o.ng ´u.ng, do`ng) cu’a no´ trong Km (tu.o.ng ´u.ng, Kn).
T`u. 2 d¯i.nh ly´ trˆen ta suy ra mˆo.t ca´ch d¯ˆe’ xe´t tı´nh d¯ˆo.c lˆa.p tuyˆe´n tı´nh hay phu. thuˆo.c tuyˆe´n tı´nh cu˜ng nhu. tı`m ha.ng cu’a mˆo.t hˆe. vector trong Kn la` d¯i tı`m ha.ng cu’a ma trˆa.n d¯u.o..c ta.o nˆen bo.’i ca´c vector d¯o´.
Vı´ du.. Xe´ t tı´nh d¯ˆo.c lˆa.p tuyˆe´n tı´nh hay phu. thuˆo.c tuyˆe´n tı´nh va` tı`m ha.ng cu’a ca´ c hˆe. vector sau:
3.5. Co. so.˙’ - Sˆo´ chiˆ` u - To.a d¯ˆo. cu˙’a khˆong gian vector.e 57 (1) (u1 = (1,1,0), u2 = (0,1,1), u3 = (1,0,1)) trong R3. Vı` 1 1 0 0 1 1 1 0 1
= 2 6= 0 nˆen (u1, u2, u3) d¯ˆo.c lˆa.p tuyˆe´n tı´nh va` rank(u1, u2, u3) = 3.
(2) (v1 = (1,1,0,0), v2 = (0,1,1,0), v3 = (2,3,1,0)) trong R4.
Lˆa.p ma trˆa.n nhˆa.n v1, v2, v3 la` ca´ c do`ng rˆo` i biˆe´n d¯ˆo’i so. cˆa´p ta d¯u.o..c:
V = 1 1 0 0 0 1 1 0 2 3 1 0 d3→d3−2d1 −−−−−−→ 1 1 0 0 0 1 1 0 0 1 1 0 d3→d3−d2 −−−−−−→ 1 1 0 0 0 1 1 0 0 0 0 0
Nhu. vˆa.y rank(V) = 2 < 3 (sˆo´ vector cu’a hˆe.), do d¯o´ (v1, v2, v3) phu. thuˆo.c tuyˆe´n tı´nh trong R4 va` rank(v1, v2, v3) = 2.
3.5 Co. so.˙’ - Sˆo´ chiˆ` u - To.a d¯ˆo. cu˙’a khˆong gian vector.e
3.5.1 Co. so.˙’ cu˙’a khˆong gian vector.
D- i.nh nghı˜a 3.7. Cho K− khˆong gian vector V. Hˆe. vector = = (e1, e2, ..., en) trong V d¯u.o.. c go.i la` mˆo.t co. so.’ cu’a V nˆe´u = d¯ˆo.c lˆa.p tuyˆe´n tı´nh va` mo.i vector cu’a V d¯ˆe` u biˆe’u thi. tuyˆe´n tı´nh qua =.
Nhˆa.n xe´ t.
(1) V´o.i mˆo.t khˆong gian vector bˆa´t ky` bao gi`o. cu˜ng tˆo` n ta.i mˆo.t co. so.’ cu’a no´ .
(2) Co. so.’ cu’a mˆo.t khˆong gian vector la` khˆong duy nhˆa´t. Va`i vı´ du..
(1) Trong K− khˆong gian vector K3 cho hˆe. gˆo` m 3 vector :
(e) = {−→e1 = (1,0,0),−→e2 = (0,1,0),−→e3 = (0,0,1)}.
Dˆe˜ thˆa´y (e) d¯ˆo.c lˆa.p tuyˆe´n tı´nh. M˘a.t kha´c:
∀−→x = (x1, x2, x3) ∈ K3 ta co´ −→x =x1−→e1 +x2−→e2 +x3−→e3.
Vˆa.y (e) la` mˆo.t co. so.’ cu’a K3.
(2) Trong K− khˆong gian vector K3 cho hˆe. gˆo` m 3 vector :
Ta co´ λ1−→u1 +λ2−→u2 +λ3−→u3 =−→0 ⇔(λ1+λ2+λ3, λ1+λ2, λ1) = −→0 ⇔ λ1+λ2+λ3 = 0 λ1+λ2 = 0 λ1 = 0 ⇔ λ1 = 0 λ2 = 0 λ3 = 0 Hay (u) d¯ˆo.c lˆa.p tuyˆe´n tı´nh.
∀−→x = (x1, x2, x3) ∈ K3, −→x = λ1−→u1 +λ2→−u2+λ3−→u3 ⇔ λ1+λ2+λ3 =x1 λ1+λ2 =x2 λ1 = x3 ⇔ λ1 =x3 λ2 =x2−x3 λ3 =x1−x2
D- iˆe` u na`y ch´u.ng to’ ∀−→x ∈ K3,−→x biˆe’u thi. tuyˆe´n tı´nh d¯u.o..c qua (u). Vˆa.y (u) la` mˆo.t co.’ so.’ cu’a K3.
(3) Trong K− khˆong gian vector Kn, hˆe. vector
(e) ={e1 = (1,0,0, ...,0,0), e2 = (0,1,0, ...,0,0), ..., en = (0,0,0, ...,0,1)}
la` mˆo.t co.’ so.’ va` co. so.’ na`y co`n d¯u.o..c go.i la` co.’ so.’ chuˆa’n t˘a´c cu’a Kn. (4) Trong K− khˆong gian vector K[x], hˆe. vector (e) ={1, x, x2, x3, ...} la` mˆo.t
co. so.’ .
3.5.2 Hˆe. sinh cu˙’a mˆo.t khˆong gian vector.
D- i.nh nghı˜a 3.8. Cho V la` mˆo.t K− khˆong gian vector, (S) la` mˆo.t hˆe. vector trong V. (S) d¯u.o.. c go.i la` hˆe. sinh cu’a khˆong gian vector V nˆe´u mo.i x ∈ V, x
bao gi`o. cu˜ ng biˆe’u thi. tuyˆe´n tı´nh d¯u.o..c qua hˆe. (S) d¯o´ . Nhˆa.n xe´ t.
(1) Mˆo.t co. so.’ cu’a K− khˆong gian vector V la` mˆo.t hˆe. sinh nhu.ng d¯iˆe` u ngu.o..c la.i khˆong d¯u´ng.
(2) Trong K− khˆong gian vector V, mˆo.t hˆe. vector la` mˆo.t co. so.’ cu’a V khi va` chı’ khi hˆe. d¯o´ d¯ˆo.c lˆa.p tuyˆe´n va` la` mˆo.t hˆe. sinh .
D- i.nh ly´ 3.6. Trong K− khˆong gian vector V cho mˆo.t hˆe. = gˆ` m h˜u.u ha.n ca´co vector (= co´ thˆe’ rˆo˜ng). Khi d¯o´ tˆa.p ho.. p tˆa´t ca’ ca´ c tˆo’ ho.. p tuyˆe´n tı´nh cu’a hˆe.
= la` mˆo.t khˆong gian cu’a V va` d¯u.o.. c go.i la` khˆong gian con sinh bo.’ i hˆe. =, kı´ hiˆe.u la` <= >.
3.5. Co. so.˙’ - Sˆo´ chiˆ` u - To.a d¯ˆo. cu˙’a khˆong gian vector.e 59
Ch´u.ng minh. D- ˘a.t W la` tˆa.p ho..p tˆa´t ca’ ca´c tˆo’ ho..p tuyˆe´n tı´nh cu’a hˆe. =. * Khi = = ∅, theo quy u.´o.c W ={0}, do d¯o´ d¯u.o.ng nhiˆen W la` mˆo.t khˆong gian con cu’a V.
* Khi = 6= ∅ va` = = (s1, s2, ..., sm) gˆo` m m vector (m ≥ 1). Khi d¯o´ , v´o.i mo.i x, y ∈ W, d¯ˆe` u tˆo` n ta.i λ1, λ2, ..., λm;µ1, µ2, ..., µm ∈ K d¯ˆe’ x =
m P i=1 λisi va` y = m P i=1 µisi. Do d¯o´ x+y = m P i=1 λisi+ m P i=1 µisi = m P i=1 (λi+µi)si ∈ W. Co`n v´o.i mo.i λ∈ K cu˜ ng co´ λx =λ
m P i=1 λisi = m P i=1 (λλi)si ∈ W.
D- u.o.ng nhiˆen, 0V ∈ W, t´u.c la` W 6= ∅. Vˆa.y W la` mˆo.t khˆong gian con cu’a
V.
Vı´ du.. Trong R3 tı`m khˆong gian con sinh bo.’ i hˆe. vector sau: (u) = {u1 = (1,1,1), u2 = (2,3,4), u3 = (4,5,6)}.
Gia’i. Lˆa.p ma trˆa.n t`u. ba do`ng u1, u2, u3