Bai Tap Khong Gian Euclide

Slide 1

KHNG GIAN EUCLIDETRN NGC DIMTch v hng v kg Euclidef l tch v hng trn kg vector V, nu:

K hiu:

Khng gian vector vi 1 tvh gi l kg Euclide.Tch v hng v kg Euclide

nh ngha:: di vector x: khong cch gia x, y

: l gc gia x v yTch v hng v khng gian EuclideTrn R2, vi tvh = 2x1y1 x1y2 x2y1 + x2y2Tnh vi x = (1,2), y = (-2,1)Tnh khong cch gia x v y Tm di vector xTrn R3 tch v hng (vi x = (x1,x2,x3), y = (y1,y2,y3))

Tnh tch ca x = (1,2,3) v y = (1,-1,2)Tnh di ca xTnh khong cch gia x, yS trc giaox, y trc giao x y = 0, S trc giao S gm cc vector i mt trc giao.S trc chun nu S trc giao v x= 1, x Siv) x M x y , yMv) M M x y , xM, yM vi) B trc giao ca M : M = {x V: x M}vii) U, W E, UW : U+W=U W: tng trc giaoS trc giaoMt s kt qu cn nh:U E, < S > = U, x U x S3. = U, < S> = U, U U S Sxy, xz x y + z, , R4. M E M ENu M E th dimM + dimM = dimV v M M = E1. x E x= 0S trc giaoMt h trc giao khng c vector 0 th c lp tuyn tnhHnh chiu trc giao:

y =prU x : hnh chiu trc giao (vung gc) ca x ln US trc giaoS ={ e1, e2,,en} l c s trc chun ca E

S trc giaoTrn R2, vi tvh = 2x1y1 x1y2 x2y1 + x2y2Vector no sau y trc giao vi nhau:x = (-1,2), y = (1,2), z = (1,1), t = (3,4)Tm 1 h trc chun t cc vector trc giao va tm cTrn R2 vi tvh chnh tc cho u=(1, -2, 1), v=(4,m+2,-1)Tm m u v v trc giao.

Lm li vi tvh sau:S trc giao3. Trn khng gian R3 vi tvh chnh tc, cho

a. Vector no sau y vung gc vi U:

b. Tm m v = ( 3, m, m 3) vung gc vi U

Lm li vi tvh: S trc giao4. Trong R3, vi tvh chnh tc cho

Tm vector u trong U sao cho u vung gc vi WS trc giao5. Trn R3 vi tvh chnh tc, tm c s ca Wa. Cho W= |

b. W l khng gian nghim ca h ptS trc giao6. Trn R3, cho 2 khg gian con

Chng minh

S trc giao7. Trong R4, cho

Tm m, n

S trc giao8. Trong R3 cho 2 kg con

Tm m

S trc giao9. Trn khng gian R3 cho S = {(1,1,1), (-2,1,1), (0,-1,1)}.Kim tra tnh trc giao ca STm 1 c s trc chun S ca R3 t S.Cho u = (1,2,2), tm ta ca u theo SS trc giaoQua trnh trc giao ha Gram - Schmidt: cho {x1, , xp} l h ltt trong E.t:

Khi {y1, , yp} l h trc giao.S trc giaoTrn khng gian R3, trc giao ha cc h vecor sau:

B sung vo cc tp hp sau c 1 c s trc giao ca R3.

S trc giaoB sung vo cc tp hp sau c 1 c s trc giao ca R4.

Cho U = , x = (-1,1,2). Tm y U, z U sao cho x = y + zS trc giaoTm hnh chiu trc giao ca

ln kg con

Trn kg R3 vi tch v hngTm hnh chiu trc giao ca

ln kg con

S trc giao

Trong R4 cho U l khng gian nghim ca h phng trnh thun nht sau:

V vector Tm hnh chiu ca z xung khng gian