由于1,n,,线性无关所以K3|=0 k1x1+k2x2+…+k k2x+k2x2+…+knxm=0 kIx,+k,,x2 0 1x1+k2x2 由于R(K)=m,则(1)式等价于下列方程组 kurr+kux K, x=0 k21x1+k2 由于 0 kmIx,+km, x2 k 0 所以方程组只有零解x1=x2 xm=0.所以v2,v2…,v线性无关 12设={x=(x1x2x,)|x1x2xn∈R且x1+x2+,+xn=0} V2={x=(x1x2xn)|x2xn∈R且x+x2++xn=1},问Ⅵ、h2是否为向量空间? 为什么? 解:集合V成为向量空间只需满足条件 V,P +B ,A∈R→Aa∈V V是向量空间,因为 0 y=(yn1,y2…,yn)y1+y2 yn 0 y=(x,x2…,xn)+(y1,y2…,yn)=(x1+y yu) 且(x1+y1)+(x2+y2)+…+(xn+yn) =(x1+x2+…+xn)+(y1+y2+…+yn)=0,故x+y∈V1 λ∈R,x=(x,x2…,xn),λx=(λx,x2,…,λxn) λx1+Ax2+…+λxn=A(x1+x2+…+xn)=0,故Ax∈V⬅Ѣ 1 2 , ,..., uu ut 㒓ᗻ᮴݇,᠔ҹ 1 2 m x x x æ ö ç ÷ × = ç ÷ ç ÷ ç ÷ è ø K o M े 11 1 12 2 1 21 1 22 2 2 11 2 2 11 2 2 0 0 0 0 m m m m r r rm m t t tm m kx kx k x kx kx k x kx kx k x kx kx k x ì + ++ = ï + ++ = ï ï í + ++ = ï ï ï + ++ = î L L LLLLLLLLLLLL L LLLLLLLLLLLL L ˄1˅ ⬅Ѣ R( ) K = m ,߭(1)ᓣㄝӋѢϟ߫ᮍ⿟㒘: 11 1 12 2 1 21 1 22 2 2 11 2 2 0 0 0 m m m m m m mm m kx kx k x kx kx k x kx kx k x ì + ++ = ïï + ++ = í ï ïî + ++ = L L LLLLLLLLLLLL L ⬅Ѣ 11 12 1 21 22 2 1 2 0 m m m m mm kk k kk k kk k ¹ L L MM M L ᠔ҹᮍ⿟㒘া᳝䳊㾷 1 2 0 m xx x === = L .᠔ҹ 1 2 , ,..., m vv v 㒓ᗻ᮴݇. 12.䆒 T 1 12 12 1 2 { ( , ,..., ) | , ,..., + +...+ 0} V xx x xx x x x x = Î= x= R nn n Ϩ T 2 12 12 1 2 { ( , ,..., ) | , ,..., + +...+ 1} V xx x xx x x x x = Î= x= R nn n Ϩ ˈ䯂V1 ǃV2ᰃ৺Ў৥䞣ぎ䯈˛ ЎҔМ˛ 㾷˖䲚ড় V ៤Ў৥䞣ぎ䯈া䳔⒵䎇ᴵӊ˖ "Î Î Þ Î "Î Î Þ Î Į V, + ȕ V Į ȕ V˗ Į V R ˈl lĮ V V1ᰃ৥䞣ぎ䯈ˈ಴Ў˖ T 12 1 2 (, , , ) 0 n n x = +++ = xx x x x x L L T 12 1 2 (, , , ) 0 n n y = + ++ = yy y y y y L L TT T 12 12 1 12 2 (, , , ) (, , , ) ( , , , ) n n nn x y += + = + + + xx x yy y x yx y x y LL L Ϩ 11 2 2 ( )( ) ( ) n n xy xy xy + + + ++ + L 12 12 ( )( )0 n n = +++ + + ++ = xx x yy y L L ˈᬙ + Î 1 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 ( )0 n n ll l l x x x xx x + ++ = +++ = L L ˈᬙl Î 1 x V