thi gia k i s tuyn tnh

K II nm hc 2014-2015

(L c Vit)

Thi gian: 3 tit

Bi 1: nh ngha ma trn ca nh x tuyn tnh?

Bi 2: nh ngha khng gian nhn v khng gian nh?

Bi 3: Chng minh khng gian nhn v khng gian nhn l cc khng gian con.

Bi 4: Cho T(x,y,z,t) = ( x + 2y + 3z +4t, y +2z +3t, x +3y +5z +7t, x+ y +z +t) . Xc nh mt c s ca Im(m),

Ker(T)

Bi 5 : Cho a1 =(1,0,0) a2 = (1,1,0) a3 = (1,1,1) ; b1 = (1,0,1) b2 = (3,1,4) b3 = (6,3,10) . T l ton t tuyn

tnh sao cho T(ai) = bi . Tm ma trn ca T ?

Bi 6 : Cho T l ton t tuyn tnh v {ui} l mt c s. Chng minh nu {T(ui)} c lp tuyn tnh th

Ker(T) = .

Bi 7 : Trong R2 nh ngha php cng v php nhn nh sau :

(x, y) + (u,v) = (x+u, y.v)

a(x, y) = (a.x, y1/a)

Kim tra 10 tnh cht ca khng gian vector.

Bi 8: Nu ba nh ngha ca c s.

Bi 9: Cho v, u l hai khng gian con dim(u) > dim (v), dim(u + v) = dim(u v) + 1 . Chng minh u+ v = u,

u +v = v .

Bi 10 : Chng minh cc ma trn i xng 33 to thnh khng gian con, tm c s ca khng gian

Bi 11 : Chng minh Tr(AB) = Tr(BA) vi Tr l vt ca ma trn.

Bi 12 : Tnh nh thc ma trn A :

A =

(

1 2 2 24 6 5 5 4 5 6 54 5 5 6)

Cu 13 : A = [0 1 01 0 01 1 1

] xc nh tr ring, vector ring ca A v tnh An.

Cu 14: Cho {ui} l mt h trc chun vsaf T l ton t tuyn tnh cho x,y l hai vector

a, Chng minh {ui} c lp tuyn tnh.

b, Chng minh nu {T(ui)} trc chun th T l ton t trc giao.

c, Chng minh = < , >< , >