显然存在矩阵A’B使得 A b2|=B B2 b +b B 所以A+B +b2 b +B 2 b b 因此R(A+B)≤R(4)+R(B) 1l.设向量组A:v,V2,…vn能由向量组B:1,l2,L1线性表示为 (v1,V2,…,Vn)=(u1,2 其中K为t×m阶矩阵,且向量组B线性无关。证明向量组A线性无关的充要条件是矩阵 K的秩R(K)=m 证明:必要性:若A组线性无关 令A=(12…,v)B=(12…,1)则有A=BK 由定理知R(K)≥R(A) 由A组:v1,V2…,v线性无关知R(A)=m,故R()≥R(A)=m 又知K为t×m阶矩阵,则R(K)≤min{t,m} 由于向量组A:v,V2…,v能由向量组B:u1,L2…,L1线性表示则m≤t mint, m)=m 综上所述知m≤R(K)≤m即R(K)=m 充分性:若R(K)= 令x"+x2"2+…+xnvn=0,其中x为实数i=1,2,…,m 则有(v1,V2,…,"n 又(",…vn)=(a1…,n)K,则(1…n)k
ᰒ✊,ᄬⶽ䰉 A¢,B¢Փᕫ T TT T 1 11 1 T TT T 2 22 2 T TT T , n ns r æö æö æ ö æ ö ç÷ ç÷ ç ÷ ç ÷ = = ¢ ¢ ç ÷ ç ÷ ç ÷ èø èø è ø è ø a b Į ȕ a b Į ȕ A B a b Į ȕ M MM M ᠔ҹ TT T T T T 11 1 1 1 1 TT T T T T 22 2 2 2 2 TT T T T T nn n n s r æ öæ öæ ö æ ö + æ ö ç ÷ç ÷ç ÷ ç ÷ ç ÷ + += = + = + ¢ ¢ ç ÷ ç ÷ ç ÷ è øè øè ø è ø + è ø ab a b Į ȕ ab a b Į ȕ AB A B ab a b Į ȕ M MM M M ℸ R(A+B)İR(A)+R(B) 11. 䆒䞣㒘 A˖ 1 2 , ,..., m vv v 㛑⬅䞣㒘 B˖ 1 2 , ,..., uu ut 㒓ᗻ㸼⼎Ў 12 1 2 ( , ,..., )=( , ,..., ) m t vv v uu uK ݊Ё KЎ t m´ 䰊ⶽ䰉ˈϨ䞣㒘 B 㒓ᗻ᮴݇DŽ䆕ᯢ䞣㒘 A 㒓ᗻ᮴݇ⱘܙ㽕ᴵӊᰃⶽ䰉 K ⱘ⾽ R m ( ) K = 䆕ᯢ˖ᖙ㽕ᗻ˖㢹 A 㒘㒓ᗻ᮴݇ Ҹ 1 1 (, , ) (, , ) Av v Bu u = = L L m t ᳝߭ A BK = ⬅ᅮ⧚ⶹ R(K)ıR(A) ⬅ A 㒘: 1 2 , ,..., m vv v 㒓ᗻ᮴݇ⶹ R(A)˙mˈᬙ R(K)ıR(A)˙m. জⶹ K Ўt m´ 䰊ⶽ䰉ˈ߭ R( ) min{ , } K tm £ ⬅Ѣ䞣㒘 A˖ 1 2 , ,..., m vv v 㛑⬅䞣㒘 B˖ 1 2 , ,..., uu ut 㒓ᗻ㸼⼎,߭ m t £ \ = min{ , } tm m 㓐Ϟ᠔䗄ⶹ m m £ £ R( ) K े R( ) K = m ˊ ߚܙᗻ˖㢹 R(K)˙m Ҹ 11 2 2 ... m m xx x v v vo + ++ = ,݊Ё i x Ўᅲ᭄i m =1,2, , L ᳝߭ 1 1 2 (, , , ) m m x x æ ö ç ÷ = ç ÷ è ø vv v o L M জ 1 1 (, , ) (, , ) v v u uK L L m t = ,߭ 1 1 (, ,)t m x x æ ö ç ÷ = ç ÷ è ø u uK o L M
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ˈЎ˖ 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