§1 KHONG GIAN VEC TO VA ANH XA TUYEN TINI
2.1. GM sU V, W Ian hurt 1a cac kheng gian sinh bai A„ ka B 1 , ..., Dr Veli mei = 1, 2, n thi B, = a,,A, + a 2A2 + u„,A„.. Do do W c V. De chUng mini) W = V, ta chi can char minh dim W dim V.
Ta c6 dim W = rang (X . X') = n — d, trong do d bang chigu cua khong gian H the nghiem ena phudng trinh:
'a 0 . 0 '
0 0 .
to p 0
t„). (X .Xt) = (0, 0, .., 0). Neat Y = (t,, t) E H Y. XX' = 0
X.)4' . = 0o Y . X. X' . = (Y.X) . (Y.X)' = 0
Y.X = 0. Nhtt vay Y e M la khOng gian nghiem cim phuong trinh t„) . X = 0 to do dim W=n—cln— dim M = rang X = dim V.
Do do W = V.
2.2. Gia su A„ ..., la the vec to cot. (-Ala ma tran X hang r. C6 th6 gia thi6t A„ (lac lap tuy6n tinh. Khi do cac Ai ; r < 5 n bie'u thi tuyett tinh qua A„ ..., A,.
Al = a,,A, + cii2A2 + + a„ A,
N611 dung ki hieu (A„ A2, ..., A,) (16 chi ma tran co cac vec to cot la A l , A2, A„ thi
aciAl)
Mei ma tran 6 v6phili c6 hang 1. 'Pa co digu phai cheing minh. 2.4. b) Xet R [x] la khong gian vec td the da thew sr& he s6 thitc. Xet u: R [x] —> R[x]
f(x) --) x .
116 rang u la chin caM, nhting khong than eau, vi Imu khong chiia nhiing da theic bac khOng.
2.5. a) Theo each the dinh caa u thi trong ea so tu nhien (e l , ..., en} cern C", to ca u(e) = e,„„ Vay A = (ad la ma trait cua u trong co so {e„ thi
{1 i (j) auo =
0 voi # a (j) b) Gia sii B = e Mat (n, C)
Xem B IA ma tram cila phep bien den tuygn tinh trong cd to nhien cna e n : v(e i )= Eb na . Khi do AB = BA a uV = vu < uv(ej) = vu(a) j = 1, 2, in n Eb u(a)= v(e a0)), j = 1, n <
n n n
Ebijecro) = Ebi,wei = <=> b ;,= bc,(;) „„, i=1 i=1
1, ..., n, nhu vay, the ma tran giao hoan dime vo tran B = (130 sao cho b u = bum on) voi myi i, j = 1, 2, .
2.6. a) De dang.
11) Gia stl u e End s (E) giao hoan voi
End K (E). Theo a) u (e) = p ie, voi {e„ cd so Gia sV i # j, 1 g 3, j < n to hay chfing minh 13, = End K (E) sao cho v(e k) = Vk 1, 2, ..., n. Tit vou vou(e) = uov(e,). Nhu vay j3, e3 = u (a) = rija a (D i P: -pj=A. Tit dou(x)=Axvoimoixe E.
2.7. Ta co the bieu din n clang (laic Ea nu jet jet n) &MI dang clang (Mk ma tran sant
an1 ant ann
( ail 312 aIn ‘ Ill 1
[a21 a22 v2 / n , . . v Vn voi mci i, j i A la cic m n. d6ng eau v nao do dm I p i . Clam v = uov say -13i) = 0 = i=