VI ma Han A=(a n) khong suy men, nen co ma tran nghich dge 11 = A ' = (bd.
EMat(2, K) Hay chUng to c6 P=
x y e Mat (2, K), det P t 0 de XP = P.Xt. z t bz = by (a-d)y= cx - bt -d)z = cx - bt cy= cz , xt zy (1)
Xet cac trudng hop sau:
+) b = c = 0, chon y = z = 0, x t = 1 +) a = d, chon y = z = 1 x = t = 0 +) b 2 +cz> 0 va a t d: chAng ban b s 0, chon x = 0, t = 1,y= b -z.
a -d
Nhtt gay trong moi trUdng hop, hP (1) dgu c6 nghigm 2.20. a) Tit u 2 (Id - u) = 0 suy ra 112 =
Tit u (Id - u)2 = 0 suy ra u - u 2 - u2 + = 0 W vgy uz = u.
b) Vi du V = K3 , u (x i , x2, x3) = (x„ x 3, 0) Ta thSy u2 * u nhung u 2 =u3 .
e) V = K3, v(x„ x2, x3) = ( 0 , x2 - x3, xa) Ta có v 2 * v nhting v (Id - v) 2 = 0.
Chu Y: Co the neu nhieu vi du khac nfia.
2.21. a) Ngu g(x) = 0 voi moi x c V thi f(x) = 0 \raj moi x e V va f = a . g voi a e K Neu có xo e V ma g(x0)* 0.
Data-
f(x) . g(x{,)
Ta chUng minh f(x) = a g (x) voi moi x c V. Vi g(xo)* 0 nen g (x) = g(x), khi do: g(x -;xo) = 0 f (x - )( 0) = f(x) - 4f(x0) = 0
f(x) = 4 . 1(x0) = g(x) f(x ) = a.g(x). g(x 0 )
b) HD: chtlng minh quy nap theo n. n = 1 dilng vi day la trUang help a).
Gia eu bai town dung voi k = 1, 2, .. n - 1. Ta chUng minh dung voi n.
Trubng help 1: Neu co i de fl Kerf c Ker f, thi fl Kerfj =
joi j#i
fl Kerf, c Kerf. Nhu vay theo gia thief quy nap, to co
i=1
f= ifJ=Eaifi (ai= 0)
Truang kip 2: Vol mdi i, n Ker fi ¢ Ker j#i
Khi do vdi mdi i (i = 1, 2, ..., n) co xi e V de? f(xi) # Ova fi(x,) = 0 ( # i). D'.1t a, -
f(x ) f
Ta chting minh dude f = + + a„ f„.
2.23. Ta chUng minh fl Kerf, 101.
Hay thong minh quy nap rang dim ( n Redd n - 1 vat i=1
1 < t < n.
TU do suy ra ket qua: Alai he phudng trinh tuyen tinh thuan nhdt vdi se; pinmng trinh ft hon s5 do dgu co nghiem khan khong.
2.25. a) De' clang
b) HD: xet 1x 1 , x„) la cd sa cua S, {x„ x„„ x,„„, xn} la co se) clia V. Voi moi i = m+1, n, set E V * : ii(Xi) = — ki hieu Kronecker). Ta thdy f, E S° . (adsbygoogle = window.adsbygoogle || []).push({});
Hay chUng minh {f„,,,, f„} is co so cfm TU do dim S ° = n - m.
2.26. a) Hidn nhien
b) Theo bai 2.25, S° IA khOng gian con cent V*, (S°)° la khong gian con dm V. Vdi x e S, fin, F e S ° , to co f(x) = 0 x e De cluing minh sinh hal S. Ta gia su W la khong gian con cna V, S c W. Ta ehling to (S°)° c W. Vdi x EW3f c V* de f(w) = 0 va f(x) x 0; Khi do vi S c W nen f(S) = 0 f
Nhung f(x) ± 0 nen x e (5 °)° . Nhit vay vdi x e W x (S°)° tit do (S°) ° c W va nbut vay (S°) ° la khong gian sinh bai S.
c) Ta cd (TY la khOng gian con cim V* va T c (T°)°.
Gia sa W* la khong gian con cim V* va W* = T.
Ta °hang to (r) ° c W.
Gia sit (fi , f„,} la co sa cim W. Neu f e V* \ W* thi f hong la t8 hop tuyen tinh cim cac LI. Do do, theo hal 21, cO x € V (x) = 0(i= 1, ..., m) nhung f(x)t O. Vi {f} la co V1?'" (x) = 0 vdi moi i = 1, 2, ... m nen g(x) = 0 vdi moi g e via do do, vdi moi g c T (vi T c W*). Tit do x e T°, Nhung f(x)# 0 f e (T°)° .
Chiang minh tren chiing to (T°)° c W*, nghia la (T°)') la khong gian con cim V*, sinh bai T.
d) HD. Gia sit (x i) c i la co sa cim khong gian V, I — tap vo ban; vdi i e I, Kat f, e V*: fi(xj) = S i (Su — ki hieu Kronecker).
X6t T la khong gian con cim V*, sinh bai (fJ ; c 1.
Hay chUng minh (T°)° = V* nhung T x V. (vi (In f E V* ma f(x) = 1 Niel moi i f 0 T).
2.27. a) va b) (16. clang
c) chi can thing minh netu V„ V2 la hai khong gian con cim V, thi (V 1 n vo° = vi° + v2"
Ta cO V, n V2 C VI, V2 VI °
v,0 c (NT, n v2)°.
D6 chiing minh (V 1 fl V2 )° c V 14) , lay f e (V1 n V2)°
f(x) = 0 vol mot x E v, n V2 . Ta cluing minh f e VI ° +
V20. )(et WI la khong gian con bit maa V, id V2 trong V, o V, = WI ED (VI n V2); Wong ta vet W2 sao cho: V2 = W2 $ (V, n V2) via V = (V, n V2) e W. Khi do V la teng trip tip gun. V, n V2,
WI, W2, W.
Chon g, h e V* the dinh bat: lb nab x E Vt n V2 hoitc X E WI
.if(x)ngu x e W2 boas x e W
t0 ngu x e VI I)V., hoac x e W2 boas x e W. f(x)ngu x e WI
WO rang g e V
1 0, he V2° va f = h + g.
Nhu vgy
(VI n V2)" t-_- W 1° + V2°. Tit dO (V, n V2)(' = Vi ± 1.72°- d) Vi du cluing to khAng dinh b) khong con dung ngu I v6 han va V co chigu ve han.
Gia sa (e) i e I la co so v6 han phis to caa V, mOt phAln to
caa V co dang Ecc,e, trong do a i hau hgt bang khOng, tit mot jel
so hilt! han. (adsbygoogle = window.adsbygoogle || []).push({});
Vol mOi i e I, ky hiQu V, la khong gian COD cila V gam cac
phan tit E cgs, vgi al = 0. ieI