V2不是向量空间,因为 r=(xu,x x1+x,+…+x.=1 y1+y2 (x1+y)+( =(x1+x2+…+xn)+(1+y2+…+yn)=1+1=2,故x+ygV2 λ∈R,x=(x1,x2…,x),Ax=(4x,Ax2…,Axn) x+x2+…+xn=(x1+x2+…+xn)=元.1=,故AxgV2 13.试证明由v1=(0,1,1),v2=(1,0,1),v3=(10)所生成的向量空间正是R 证明:设A=(V,V2,v3) v2 01 于是R(43故V,V2,V3线性无关。由于v,V2,3均为三维,且秩为3,所以v,V2,"3为此 三维空间的一组基,故由v1,n2,v2所生成的向量空间就是R3 14.由v1=(1,1,0,0)},v2=(1,0,1,1)所生成的向量空间记为,由 v3=(2,-13,3),v4=(0,12-1,-1)所生成的向量空间记为2,证明:1=V2 证明:设={x=kn+k21,k∈R,={=An+1,2∈R 由于v,V2线性无关,v3,v4也线性无关,所以,v2、"3,V分别是向量空间H、V2的 组基。又 10-110-1-31013-1 013-1013-10000 10-11 013 31 0000 0000 0000 0000V2 ϡᰃ৥䞣ぎ䯈ˈ಴Ў˖ T 12 1 2 (, , , ) 1 n n x = +++ = xx x x x x L L T 12 1 2 (, , , ) 1 n n y = + ++ = yy y y y y L L 11 2 2 ( )( ) ( ) n n xy xy xy + + + ++ + L 12 12 ( ) ( ) 11 2 n n = + + + + + + + =+= xx x yy y L L ˈᬙ + Ï 2 x yV T T 12 1 2 , (, , , ) ( , , , ) n n l l ll l Î = Rx x xx x x x x L L ˈ = 1 2 12 ( )1 n n ll l l l l x x x xx x + + + = + + + = ×= L L ˈᬙl Ï 2 x V 13ˊ䆩䆕ᯢ⬅ TTT 123 vvv === (0,1,1) (1,0,1) (1,1,0) ˈ ˈ ᠔⫳៤ⱘ৥䞣ぎ䯈ℷᰃ R 3 . 䆕ᯢ˖䆒 123 A vvv = (, , ) 1 123 011 110 , , 1 0 1 ( 1) 1 0 1 2 0 110 011 - A vvv = = = - =- ¹ Ѣᰃ R(A)=3 ᬙ 123 vvv , , 㒓ᗻ᮴݇DŽ⬅Ѣ 123 vvv , , ഛЎϝ㓈ˈϨ⾽Ў 3ˈ᠔ҹ 123 vvv , , Ўℸ ϝ㓈ぎ䯈ⱘϔ㒘෎ˈᬙ⬅ 123 vvv , , ᠔⫳៤ⱘ৥䞣ぎ䯈ህᰃ 3 R DŽ 14ˊ⬅ T T 1 2 v v = = (1,1,0,0) , (1,0,1,1) ᠔⫳៤ⱘ৥䞣ぎ䯈䆄Ў V1ˈ⬅ T T 3 4 v v = - = -- (2, 1,3,3) , (0,1, 1, 1) ᠔⫳៤ⱘ৥䞣ぎ䯈䆄Ў V2ˈ䆕ᯢ˖V1 = V2 䆕ᯢ˖䆒Vxv v V yv v 1 11 2 2 1 2 2 13 2 4 1 2 == + Î == + Î { k k kk R R ,, , } { l l ll } ⬅Ѣ 1 2 v v, 㒓ᗻ᮴݇ˈ 3 4 v v, г㒓ᗻ᮴݇ˈ᠔ҹ 1 2 v v, ǃ 3 4 v v, ߿ߚᰃ৥䞣ぎ䯈V V 1 2 ǃ ⱘϔ 㒘෎DŽজ ( 1234 ) 11 2 0 1 1 2 0 112 0 10 1 1 0 1 3 1 013 1 = 01 3 1 0 1 3 1 000 0 01 3 1 0 1 3 1 000 0 æ öæ öæ ö ç ÷ç ÷ç ÷ - -- - - - è øè øè ø - - vvvv : : 1 1 1 0 10 1 1 2 2 01 3 1 3 1 0 1 00 0 0 2 2 0 000 00 0 0 0 000 æ ö - ç ÷ æ ö ç ÷ ç ÷ - ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ è ø ç ÷ è ø : :