第四次作业 1设v1=(1,0),V2=(0,1),n3=(3,4,0),求v-2及3v+2V2-v2 解:V1-"2=(,10)-(0,1,1)2=(1-01-10-1)=(10,-1 3v1+2v2-v2=3(10)2+2(0,1)-(3,4.0)2=(3-3,3+2-42)=(0,1,2) 2设3(V1-v)+2(V2+)=5(3+v),求.其中v1=(2,5,1,3)2,v2=(10,1,5,10)y, v3=(4,1,-1,1) 解:由3(V1-ν)+2(以2+v)=5(V3+ν)整理得 (31+2v2-5v2)=[B3(2,5,1,3)+2(10,1,5,10)-5(41-1,1)] =(1,2,3,4) 3选择题 1)向量组v,V2,…,v线性无关的充要条件是(C (A)v,V2,…,vn均为非零向量 (B)v,V2…,vn的任何两个向量分量成比例; (C)v,V2…,vn中任何一个向量不能被其余向量线性表示; (D)2V2…,vn中有一部分线性无关。 2),V2…,Vn为一向量组,则(c (A)若kv1+k2v2+…+knvn=0,则v,V2…,v线性无关 (B)对于任意不全为零的k,k2…,km均有k1+k2v2+…+knv=0,则v,V2…,v线性 无关 (C)v,v2…,v线性相关,则对任意不全为零的k,k2,…,k有kv1+k2v2+…+knvn=0 (D)若0v1+0V2+…+0m=0,则v,v2…,v线性无关 3)v1,v2…,vn和n1,n2…,un是两向量组,存在两组不全为零的数k1,k2kn和l1,l2…,ln 使(k1+l1+(k2+l2)2++(km+lm)vn=0,则 (A)v,v2,vn和1n2,Ln都线性无关 (B)v2V2,vn和1,B2,un都线性相关 (C)v1+1,V2+l2…,Vn+Ln,V1-u1,V2-l2…,vn-ln线性无关 (D)V+1,V2+2,…,Vm+un,1一B1,V2-42,…,Vn-Ln线性相关 4)向量组v,V2,v线性无关,则以下向量组线性无关的是() (A)v+"2,V2+"3"3+ (B)V1+v2,V2+V3,V1+2v2+v3 (C)v1+2v2,2v2+3v3,3v3+v 2v1-3v2+3v3,3v1-2v2+3v 5)若a,月,"线性无关,a,B,6线性相关,则() (A)a必可由,7,6线性表示(B)β必可由a,,6线性表示
ಯϮ 1.䆒 TT T 123 vvv === (1,1,0) (0,1,1) (3,4 0) ˈ ˈ ˈ ˈ∖ 12 1 23 vv v vv - +- ঞ3 2 㾷˖ TT T T 1 2 v v - = - =- - - = - (1,1,0) (0,1,1) (1 0,1 1,0 1) (1, 1) 0, T TT TT 1 23 3 2 3(1,1,0) 2(0,1,1) (3,4,0) (3 3,3 2 4,2) (0,1, v vv + - = + - = - +- = 2) 2.䆒 12 3 3( ) 2( ) 5( ) vv vv vv -+ += + ˈ∖ v .݊Ё T 1 v = (2,5,1,3) ˈ T 2 v = (10,1 5,10) ˈ ˈ T 3 v = (4,1, 1,1) - 㾷˖⬅ 12 3 3( ) 2( ) 5( ) vv vv vv -+ += + ᭈ⧚ᕫ T TT 123 T 1 1 (3 2 5 ) [3(2,5,1,3) 2(10,1,5,10) 5(4,1, 1,1) ] 6 6 (1,2,3,4) = +- = + - - = v vvv ǂǂǂǂǂǂǂǂǂ 3.䗝ᢽ乬 1)䞣㒘 1 2 , ,..., m vv v 㒓ᗻ᮴݇ⱘܙ㽕ᴵӊᰃ ( C ) (A) 1 2 , ,..., m vv v ഛЎ䴲䳊䞣˗ (B) 1 2 , ,..., m vv v ⱘӏԩϸϾ䞣ߚ䞣៤↨՟˗ (C) 1 2 , ,..., m vv v ЁӏԩϔϾ䞣ϡ㛑㹿݊ԭ䞣㒓ᗻ㸼⼎˗ (D) 1 2 , ,..., m vv v Ё᳝ϔ䚼ߚ㒓ᗻ᮴݇DŽ 2) 1 2 , ,..., m vv v Ўϔ䞣㒘ˈ߭ ( c ) (A)㢹 11 2 2 ... m m kk k v v v0 + ++ = ˈ߭ 1 2 , ,..., m vv v 㒓ᗻ᮴݇ (B)ᇍѢӏᛣϡܼЎ䳊ⱘ 1 2 , ,..., m kk k ഛ᳝ 11 2 2 ... m m kk k v v v0 + ++ = ˈ߭ 1 2 , ,..., m vv v 㒓ᗻ ᮴݇ (C) 1 2 , ,..., m vv v 㒓ᗻⳌ݇ˈ߭ᇍӏᛣϡܼЎ䳊ⱘ 1 2 , ,..., m kk k ᳝ 11 2 2 ... m m kk k v v v0 + ++ = (D)㢹 1 2 0 0 ... 0 + ++ = m v v v0 ˈ߭ 1 2 , ,..., m vv v 㒓ᗻ᮴݇ 3) 12 1 2 , ,..., , ,..., m m vv v uu u ᰃϸ䞣㒘ˈᄬϸ㒘ϡܼЎ䳊ⱘ᭄ 1 2 , ,..., m kk k 1 2 , ,..., m ll l Փ 1 11 2 2 2 ( ) ( ) ... ( ) m mm kl kl k l + + + ++ + = v v v0 ˈ߭ (A) 12 1 2 , ,..., , ,..., m m vv v uu u 䛑㒓ᗻ᮴݇ (B) 12 1 2 , ,..., , ,..., m m vv v uu u 䛑㒓ᗻⳌ݇ (C) 1 12 2 1 12 2 , ,..., , ,..., ++ -- - mm m m v uv u v u v uv u v u + , 㒓ᗻ᮴݇ (D) 1 12 2 1 12 2 , ,..., , ,..., ++ -- - mm m m v uv u v u v uv u v u + , 㒓ᗻⳌ݇ 4)䞣㒘 123 vvv , , 㒓ᗻ᮴݇ˈ߭ҹϟ䞣㒘㒓ᗻ᮴݇ⱘᰃ ( ) (A) 1 22 33 1 v vv vv v +++ , , (B) 1 2 2 31 2 3 v vv vv v v + + ++ , ,2 (C) 1 2 2 33 1 v vv vv v + ++ 2 ,2 3 ,3 (D) 1 2 31 2 31 2 3 vv vv v vv v v ++ - + - + ,2 3 3 ,3 2 3 5)㢹Į ȕ, , Ȗ 㒓ᗻ᮴݇ˈĮ ȕ, ,į 㒓ᗻⳌ݇ˈ߭ ( ) (A) Į ᖙৃ⬅ ȕ Ȗ, ,į㒓ᗻ㸼⼎ (B) ȕ ᖙৃ⬅Į Ȗ, ,į㒓ᗻ㸼⼎
(C)d必可由a,B,y线性表示D)d必不可由a,月,y线性表示 6)向量组v,v2…,V线性无关的充要条件是() (A)存在不全为零的数k1,k2…,km,使kv+k2V2+…+knvn=0 (B)v,V2…,vn中任意丙个向量都线性无关 (C)v,V2…,vn中存在一个向量,不能用其余向量线性表示 (D)v2V2…,vn中任一向量不可用其余向量线性表示 7)设v2V2,V3,v4线性无关,则() (A)V1+V2,V2+v3,v3+V4,ν4+v线性无关 (B)V-"2,V2-V3,V3-v4,V-V线性无关 (C)V+2,V2+V3,V3+V4,V4+V线性无关 (D)V1+"2,V2+v3V3-v4,V4-V线性无关 8)设A是n阶方阵,R(A)=r<n,那么在A的n个行向量中() (A)必有r个行向量线性无关 (B)任意r个行向量线性无关 (C)任意r个行向量构成大线性无关组 (D)任意一个行向量都可由其它r个行向量线性表示 9)设Amn的秩R(4)=m<n,En是m阶单位矩阵,下面结论正确的是( (A)4的任意m个列向量必线性无关 (B)A的任意m阶子式不等于零 (C)4经行的初等变换,可化为(EmO)的形式 (D)若BA=O,则B=0或方程组Ax=b有无穷多组解 10)设A为m×n阶矩阵,齐次线性方程组Ax=0仅有零解的充分必要条件是() (A)列向量线性无关 (B)列向量线性相关 (C)行向量线性无关 (D)行向量线性相关 4设码1=V1u2=V+"2…,Ⅱn=V1+v2+…+vn,且向量组v1,v2…,v线性无关,证明 向量组u1,2…,Ln也线性无关 证明:设kL1+k2L2+…+k,Ln=0则 (k+…+km)+(k2+…+km)2+…+(k,+…+km)V+…+kn"m=0 因向量组v,V2…v线性无关,故
(C) į ᖙৃ⬅Į ȕ, , Ȗ 㒓ᗻ㸼⼎ (D) į ᖙϡৃ⬅Į ȕ, , Ȗ 㒓ᗻ㸼⼎ 6)䞣㒘 1 2 , ,..., m vv v 㒓ᗻ᮴݇ⱘܙ㽕ᴵӊᰃ ( ) (A)ᄬϡܼЎ䳊ⱘ᭄ 1 2 , ,..., m kk k ˈՓ 11 2 2 ... m m kk k v v v0 + ++ = (B) 1 2 , ,..., m vv v ЁӏᛣϭϾ䞣䛑㒓ᗻ᮴݇ (C) 1 2 , ,..., m vv v ЁᄬϔϾ䞣ˈϡ㛑⫼݊ԭ䞣㒓ᗻ㸼⼎ (D) 1 2 , ,..., m vv v Ёӏϔ䞣ϡৃ⫼݊ԭ䞣㒓ᗻ㸼⼎ 7)䆒 1234 vvvv ,,, 㒓ᗻ᮴݇ˈ߭ ( ) (A) 1 22 33 44 5 v vv vv vv v ++++ ,,, 㒓ᗻ᮴݇ (B) 1 22 33 44 1 v vv vv vv v ---- ,,, 㒓ᗻ᮴݇ (C) 1 22 33 44 1 v vv vv vv v ++++ ,,, 㒓ᗻ᮴݇ (D) 1 22 33 44 1 v vv vv vv v ++-- ,,, 㒓ᗻ᮴݇ 8)䆒 A ᰃ n 䰊ᮍ䰉ˈ R rn ( ) A = < ˈ䙷М A ⱘ n Ͼ㸠䞣Ё ( ) (A)ᖙ᳝ r Ͼ㸠䞣㒓ᗻ᮴݇ (B)ӏᛣ r Ͼ㸠䞣㒓ᗻ᮴݇ (C)ӏᛣ r Ͼ㸠䞣ᵘ៤㒓ᗻ᮴݇㒘 (D)ӏᛣϔϾ㸠䞣䛑ৃ⬅݊ᅗ r Ͼ㸠䞣㒓ᗻ㸼⼎ 9)䆒 Am n´ ⱘ⾽ R mn ( ) A = < ˈ Em ᰃ m 䰊ऩԡⶽ䰉ˈϟ䴶㒧䆎ℷ⹂ⱘᰃ ( ) (A)A ⱘӏᛣ m Ͼ߫䞣ᖙ㒓ᗻ᮴݇ (B) A ⱘӏᛣ m 䰊ᄤᓣϡㄝѢ䳊 (C) A 㒣㸠ⱘ߱ㄝবᤶˈৃ࣪Ў(E O m ) ⱘᔶᓣ (D)㢹 BA O= ˈ߭ B O= ᮍ㒘 Ax b = ᳝᮴か㒘㾷 10)䆒 A Ў m×n 䰊ⶽ䰉ˈ唤㒓ᗻᮍ㒘 Ax 0 = ҙ᳝䳊㾷ⱘߚܙᖙ㽕ᴵӊᰃ˄ ˅ (A)߫䞣㒓ᗻ᮴݇ (B)߫䞣㒓ᗻⳌ݇ (C)㸠䞣㒓ᗻ᮴݇ (D)㸠䞣㒓ᗻⳌ݇ 4.䆒 1 12 1 2 1 2 , ,..., ... u vu v v u v v v = = + = + ++ m m ˈϨ䞣㒘 1 2 , ,..., m vv v 㒓ᗻ᮴݇ˈ䆕ᯢ 䞣㒘 1 2 , ,..., uu um г㒓ᗻ᮴݇ 䆕ᯢ˖䆒 11 2 2 r m kk k u u uo + ++ = L ߭ 1 12 2 ( )( ) ( ) m m p m p mm kkk k k k k ++ + ++ ++ ++ ++ = L L LL L v v v vo 䞣㒘 1 2 , ,..., m vv v 㒓ᗻ᮴݇,ᬙ
k1+k2+…+kn=0 01 0 0 01八(k 0 因为01 1≠0故方程组只有零解 则k1=k2=…=kn=0所以1,n2…,Lm线性无关 5利用初等变换求下列矩阵的列向量组的一个极大线性无关组 25311743 22 453132 759454134 203-13 1104-1 253117432(25311743 25311743 759453132 0123 解:(1) 25322048-n0135/0013 759454134 0 35 0000 所以第1、2、3列构成一个最大无关组 l1221 221 l122 0215-15+ 203-13 0-2 1104 r4-1 00000 所以第1、2、3列构成一个最大无关组 6求下列向量组的秩,并求一个极大线性无关组 1)v1=(12-14),"2=(9100104),"3=(-2-42-8) 2)n=(1213),yn2=(4 5-6),v=(1 解:(1)-2v1=v3→V1,线性相关 12-14 由v 9100104~108219-32 2-428)(0000 秩为2,一组最大线性无关组为v,V2
1 2 1 2 2 0 11 0 0 01 1 0 0 01 0 0 m m r m kk k k k k k k k ì +++ = æ ö æö æ ö ï ++ = ç ÷ ç÷ ç ÷ ï í Û = ç ÷ ç÷ ç ÷ ï ç ÷ ç ÷ ï = è ø èø è ø î L L L L L LLLLLL MLLM M M L Ў 1 1 01 1 1 0 0 01 = ¹ L L L MLLM L ᬙᮍ㒘া᳝䳊㾷 ߭ 1 2 0 m kk k === = L ᠔ҹ 1 2 , ,..., uu um 㒓ᗻ᮴݇ 5.߽߱⫼ㄝবᤶ∖ϟ߫ⶽ䰉ⱘ߫䞣㒘ⱘϔϾᵕ㒓ᗻ᮴݇㒘 1) 25 31 17 43 75 94 53 132 75 94 54 134 25 32 20 48 æ ö ç ÷ è ø 2) 112 2 1 021 5 1 203 1 3 110 4 1 æ ö ç ÷ - - è ø - 㾷˖(1) 2 1 4 3 3 1 3 2 4 1 25 31 17 43 25 31 17 43 25 31 17 43 3 75 94 53 132 0 1 2 3 0 1 2 3 75 94 54 134 0 1 3 5 0 0 1 3 3 25 32 20 48 0 1 3 5 0 0 0 0 ~ ~ r r r r r r r r r r æ ö æ öæ ö - - ç ÷ ç ÷ç ÷ - - è ø è øè ø - ᠔ҹ 1ǃ2ǃ3 ߫ᵘ៤ϔϾ᳔᮴݇㒘. (2) 3 1 32 41 3 4 112 2 1 1 1 2 2 1 11 2 2 1 2 021 5 1 0 2 1 5 1 02 1 5 1 203 1 3 0 2 1 5 1 00 22 2 110 4 1 0 0 2 2 2 00 0 0 0 ~ ~ r r rr rr r r æ öæ ö æ ö - + ç ÷ç ÷ ç ÷ - -- - --- - - - « è øè ø è ø - -- ᠔ҹ 1ǃ2ǃ3 ߫ᵘ៤ϔϾ᳔᮴݇㒘ˊ 6.∖ϟ߫䞣㒘ⱘ⾽ˈᑊ∖ϔϾᵕ㒓ᗻ᮴݇㒘 1) ( ) ( ) ( ) TT T 12 3 vv v = - = =- - - 1 2 1 4 , 9 100 10 4 , 2 4 2 8 2) ( ) ( ) ( ) TT T 12 3 vv v = = --- = --- 1 2 1 3, 4 1 5 6, 1 3 4 7 㾷˖(1) 1 3 13 - =Þ 2 , v v vv 㒓ᗻⳌ݇. ⬅ T 1 T 2 T 3 1 2 14 12 1 4 9 100 10 4 0 82 19 32 2 4 2 8 00 0 0 ~ æ ö æ öæ ö - - ç ÷ = - ç ÷ç ÷ ç ÷ ç ÷ è øè ø -- - è ø v v v ⾽Ў 2,ϔ㒘᳔㒓ᗻ᮴݇㒘Ў 1 2 v v,
918~0112 秩为2最大线性无关组为v,v 7设v,V2,…,v是一组n维向量,已知n维单位坐标向量e,e2…,en能由它们线性表示,证 明v,V2,…,vn线性无关。 证明:由于n维单位向量e1:巳2…,en线性无关,不妨设 k1v1+k12v2+…+knvn k21V1+k2V2+…+k2nv en=kn1+kn22+…+kn k1k12…k k 所以 en k, k 两边取行列式,得 由 ≠0 即n维向量组v,V2,…,n所构成矩阵的秩为n,故v1,以2…,vn线性无关 8.设v,V2…,v是一组n维向量,证明它们线性无关的充要条件是:任一n维向量都可由 它们线性表示。 证明:设e,e2,en为一组n维单位向量,对于任意n维向量a=(k,k2,…kn) 则有a=ke1+k2e2+…+ken,即任一n维向量都可由单位向量线性表示 必要性:设2V2…,V线性无关,且v,V2…,vn能由单位向量线性表示,即 V1=k1e1+k12e2+…+kne k1k12…kn h2e,+k22e,++k2n,vik2k kne1+kn2e2+…+km knl k 两边取行列式,得
(2) T 1 T 2 T 3 1 2 1 3 1 2 1 3 1213 4 1 5 6 0 9 9 18 0 1 1 2 1 3 4 7 0 5 5 10 0 0 0 0 ~ ~ æ ö æ öæ öæ ö ç ÷ = --- --- ç ÷ç ÷ç ÷ ç ÷ ç ÷ è øè øè ø --- --- è ø v v v ⾽Ў 2,᳔㒓ᗻ᮴݇㒘Ў T T 1 2 v v, . 7.䆒 1 2 , ,..., n vv v ᰃϔ㒘 n 㓈䞣ˈᏆⶹ n 㓈ऩԡതᷛ䞣 1 2 , ,..., n ee e 㛑⬅ᅗӀ㒓ᗻ㸼⼎ˈ䆕 ᯢ 1 2 , ,..., n vv v 㒓ᗻ᮴݇DŽ 䆕ᯢ˖⬅Ѣ n 㓈ऩԡ䞣 1 2 , ,..., n ee e 㒓ᗻ᮴݇ˈϡོ䆒˖ 1 11 1 12 2 1 2 21 1 22 2 2 11 2 2 n n n n n n n nn n kk k kk k kk k = + ++ = + ++ = + ++ evv v e vv v evv v L L LLLLLLLLLLLL L ᠔ҹ T T 1 1 11 12 1 T T 2 2 21 22 2 T T 1 2 n n n n nn n n kk k kk k kk k æö æö æ ö ç÷ ç÷ ç ÷ = ç ÷ ç ÷ èø èø è ø e v e v e v L L M M LLLL L ϸ䖍প㸠߫ᓣˈᕫ T T 1 1 11 12 1 T T 2 2 21 22 2 T T 1 2 n n n n nn n n kk k kk k kk k = e v e v e v L L M M LLLL L ⬅ T T 1 1 T T 2 2 T T 10 0 n n =¹Þ ¹ e v e v e v M M े n 㓈䞣㒘 1 2 , ,..., n vv v ᠔ᵘ៤ⶽ䰉ⱘ⾽Ў nˈᬙ 1 2 , ,..., n vv v 㒓ᗻ᮴݇. 8. 䆒 1 2 , ,..., n vv v ᰃϔ㒘 n 㓈䞣ˈ䆕ᯢᅗӀ㒓ᗻ᮴݇ⱘܙ㽕ᴵӊᰃ˖ӏϔ n 㓈䞣䛑ৃ⬅ ᅗӀ㒓ᗻ㸼⼎DŽ 䆕ᯢ˖䆒 1 2 , ,..., n ee e Ўϔ㒘 n 㓈ऩԡ䞣ˈᇍѢӏᛣ n 㓈䞣 T 1 2 (, , , ) n a = kk k L ˈ ᳝߭ 11 2 2 n n ae e e = + ++ kk k L ˈेӏϔ n 㓈䞣䛑ৃ⬅ऩԡ䞣㒓ᗻ㸼⼎. ᖙ㽕ᗻ˖䆒 1 2 , ,..., n vv v 㒓ᗻ᮴݇ˈϨ 1 2 , ,..., n vv v 㛑⬅ऩԡ䞣㒓ᗻ㸼⼎ˈे T T 1 11 1 12 2 1 11 12 1 1 1 T T 2 21 1 22 2 2 21 22 2 2 2 T T 11 2 2 1 2 nn n nn n n n n nn n n n nn n n k k k kk k k k k kk k k k k kk k = + ++ æö æö æ ö = + ++ ç÷ ç÷ ç ÷ Û = ç ÷ ç ÷ = + ++ èø èø è ø vee e v e v ee e v e vee e v e L L L L LLLLLLLLLLLL L L L L M M L L ϸ䖍প㸠߫ᓣˈᕫ
叫|k1k2…knle k1k12…kn k2n2|由2|≠0→ knk2…k k k k1k12…kn 令A k, k ,则矩阵A可逆 knkn2…k 由 A A 即e1:巳2…,en都能由,v2,…,v线性表示,因为任一n维向量能由单位向量e1,e2,en线 性表示,故任一n维向量都可以由v,V2…,v线性表示 充分性:已知任一n维向量都可由v1,V2,…,v线性表示,则单位向量组:e1,e2…,en可由 v1,V2,…,V线性表示,由8题知v1,V2…,v线性无关 9.设向量组A:v,V2,…,v的秩为,向量组B:1,2…,L的秩为F2,向量组C: v1,2,…,Vn,1,u2,u1的秩为,证明max{,2}≤F≤F+l2 证明:设A,BC的最大线性无关组分别为ABC,含有的向量个数(秩)分别为F,2,F,则 ABC分别与A'B'C等价易知A,B均可由C线性表示则R()≥R(4)R(O≥R(B),即 max{,n2}≤r3 设A与B中的向量共同构成向量组D,则A,B均可由D线性表示即C可由D线性表 示从而C可由D线性表示,所以R(C)≤R(D),而D中共有1+F2个向量,所以 R(D)≤F+2即乃≤F+F2 10证明R(A+B)≤R(A)+R(B) 证明:设A=(a1,a2…,an),B=(b,b2…,bn) 且A,B行向量组的最大无关组分别为a,双2,…、B1,B22…,BF
T T 1 1 11 12 1 T T 2 2 21 22 2 T T 1 2 n n n n nn n n kk k kk k kk k = v e v e v e L L M M LLLL L ⬅ T 1 11 12 1 T 2 21 22 2 T 1 2 0 0 n n n n nn n kk k kk k kk k ¹Þ ¹ v v v L L M LLLL L Ҹ 11 12 1 21 22 2 1 2 n n n n nn kk k kk k kk k æ ö ç ÷ = ç ÷ ç ÷ è ø A L L LLLL L ˈ߭ⶽ䰉 A ৃ䗚 ⬅ TT T T 11 1 1 TT T T 1 22 2 2 TT T T nn n n - æö æö æö æö ç÷ ç÷ ç÷ ç÷ = Þ= èø èø èø èø ve e v ve e v A A ve e v MM M M े 1 2 , ,..., n ee e 䛑㛑⬅ 1 2 , ,..., n vv v 㒓ᗻ㸼⼎ˈЎӏϔ n 㓈䞣㛑⬅ऩԡ䞣 1 2 , ,..., n ee e 㒓 ᗻ㸼⼎ˈᬙӏϔ n 㓈䞣䛑ৃҹ⬅ 1 2 , ,..., n vv v 㒓ᗻ㸼⼎. ߚܙᗻ˖Ꮖⶹӏϔ n 㓈䞣䛑ৃ⬅ 1 2 , ,..., n vv v 㒓ᗻ㸼⼎ˈ߭ऩԡ䞣㒘˖ 1 2 , ,..., n ee e ৃ⬅ 1 2 , ,..., n vv v 㒓ᗻ㸼⼎ˈ⬅ 8 乬ⶹ 1 2 , ,..., n vv v 㒓ᗻ᮴݇. 9.䆒䞣㒘 A˖ 1 2 , ,..., m vv v ⱘ⾽Ў 1 r ˈ䞣㒘 B˖ 1 2 , ,..., uu ut ⱘ⾽Ў 2 r ˈ䞣㒘 C˖ 12 1 2 , ,..., , , ,..., m t vv v uu u ⱘ⾽Ў 3 r ˈ䆕ᯢ max{ , } 12 3 1 2 rr r r r £ £+ 䆕ᯢ˖䆒 A,B,C ⱘ᳔㒓ᗻ᮴݇㒘߿ߚЎ A¢,B¢,C ¢,᳝ⱘ䞣Ͼ᭄(⾽)߿ߚЎ 123 rrr , , ,߭ A,B,C ߿ߚϢ A¢,B¢,C ¢ㄝӋ,ᯧⶹ A,B ഛৃ⬅ C 㒓ᗻ㸼⼎,߭ R(C)ıR(A),R(C)ıR(B)ˈे max{ , } 12 3 rr r £ 䆒 A¢Ϣ B¢Ёⱘ䞣݅ৠᵘ៤䞣㒘 D,߭ AˈB ഛৃ⬅ D 㒓ᗻ㸼⼎,े C ৃ⬅ D 㒓ᗻ㸼 ⼎,Ң㗠 C ¢ৃ⬅ D 㒓ᗻ㸼⼎ˈ᠔ҹ R(C ¢)İR(D)ˈ㗠 D Ё᳝݅ 1 2 r r + Ͼ䞣ˈ᠔ҹ R(D)İ 1 2 r r + े 312 r rr £ + . 10.䆕ᯢ R RR ( ) () () AB A B +£ + 䆕ᯢ˖䆒 T T 12 12 ( , , , ), ( , , , ) A aa a B bb b = = L L n n Ϩ AˈB 㸠䞣㒘ⱘ᳔᮴݇㒘߿ߚЎ TT T 1 2 , ,, Į Į L Įr ǃ TT T 1 2 , ,, s ȕ ȕ L ȕ
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
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 0000
V2 ϡᰃ䞣ぎ䯈ˈЎ˖ 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 : :
v3=3v2-v,v4=V1-v2 v==V3+=v4,v2==(v3+v4) 故V1=V2 15.证明v1=(1,-1,0),v2=(2,1,3),v=(3,1,2)是R的一组基,并把 a1=(5,0,7),a2=(-9,-8,-13)用该基线性表示。 1235 1235 证明:因为("2"aa2)=-1110-8-0345-17 0327-13(00 (1235-9)(10023) 0345-17-0103-3 001 所以v,P2v2线性无关,又它们是三维向量,故是R的一组基,且 16.求下列线性方程组的基础解系 x1-8x2+10x3+24=0 x4=0 (1){2x1+4x2+5 0 (2){3x1+5x2+4x3-2x4=0 3x1+8x,+6x3-2x4=0 8x1+7x2+6x3-3x4 1040 8102 初等行变换 解:(1)A=245-1 386 44 0000 所以原方程组等价于 X3 RI 0 取x3=0,x4=4得x1=0,x2=1 因此基础解系为51= 4
3 2 14 1 2 v v vv v v = - =- 3 , 1 3 42 3 4 13 1 ,() 22 2 v v vv v v =+ = + ᬙV V 1 2 = 15ˊ䆕ᯢ TTT 1 23 v vv =- = = (1, 1,0) (2,1,3) (3,1, 2) ˈ ˈ ᰃ R 3 ⱘϔ㒘ˈᑊᡞ T T 1 2 a a = =- - - (5,0,7) , ( 9, 8, 13) ⫼䆹㒓ᗻ㸼⼎DŽ 䆕ᯢ˖Ў( 12312 ) 1 235 9 12 3 5 9 1 1 1 0 8 0 3 4 5 17 0 3 2 7 13 0 0 2 2 4 æ öæ ö - - ç ÷ç ÷ =- - - è øè ø - - vvvaa : 123 5 9 100 2 3 0 3 4 5 17 0 1 0 3 3 001 1 2 001 1 2 æ öæ ö - ç ÷ç ÷ - - è øè ø - - -- : : ᠔ҹ 123 vvv , , 㒓ᗻ᮴݇ˈজᅗӀᰃϝ㓈䞣ˈᬙᰃ R 3 ⱘϔ㒘ˈϨ 1 1 23 a v vv =+- 2 3 ˈ 2 12 3 a vv v =-- 33 2 16ˊ∖ϟ߫㒓ᗻᮍ㒘ⱘ⸔㾷㋏ (1) 1 2 34 1 2 34 1234 8 10 2 0 245 0 38 6 2 0 xx x x x xx xx xx ì - + += ï í + + -= ï î ++- = (2) 1 2 34 1234 1234 23 2 0 35 42 0 87 63 0 x xx xx xx xxxx ì - - += ï í ++-= ï î ++-= 㾷˖(1) 10 4 0 1 8 10 2 3 1 2 4 5 1 01 4 4 38 6 2 00 0 0 ~ æ ö æ ö - ç ÷ = - -- ç ÷ ç ÷ ç ÷ - ç ÷ è ø è ø A ߱ㄝ㸠বᤶ ᠔ҹॳᮍ㒘ㄝӋѢ 1 3 2 34 4 3 1 4 4 x x x xx ì = - ï í = + ï î প 3 4 x x = =- 1, 3ᕫ 1 2 x x =- = 4, 0 প 3 4 x x = = 0, 4ᕫ 1 2 x x = = 0, 1 ℸ⸔㾷㋏Ў 1 2 4 0 0 1 , 1 0 3 4 æ ö æö - ç ÷ ç÷ = = ç ÷ ç÷ ç ÷ ç÷ è ø èø - ȟ ȟ
初等行变换 (2)A=354-2 011479 876-3 1919 000 XI 所以原方程组等价于 x,=--X2+-x 取x3=1,x4=2得x1=0,x2=0 取x3=0,x4=19得x1=1,x2=7 因此基础解系为51=11 1设A=(9-528,求一个4×2阶矩阵B,使AB=0,且R(B)=2 解:由于R(A=2,所以方程AX=O的解空间的秩也是2,所以矩阵B就可由该解空间中任 何两个线性无关的向量构成 而A 22 8 04 01 所以方程AX=O的基础解系为1=。,2 0 故所求矩阵B=511 80 18.求一个齐次线性方程组,使它的基础解系为51=(0,1,2,3),k2=(3,2,1,0 解:显然所求方程组的通解为
(2) 2 1 1 0 19 19 2 3 21 14 7 3 5 4 2 01 19 19 87 6 3 00 0 0 ~ æ ö - ç ÷ æ ö - - ç ÷ =- - ç ÷ ç ÷ ç ÷ ç ÷ è ø - ç ÷ ç ÷ è ø A ߱ㄝ㸠বᤶ ᠔ҹॳᮍ㒘ㄝӋѢ 1 34 2 34 2 1 19 19 14 7 19 19 x xx x xx ì =- + ïï í ï =- + ïî প 3 4 x x = = 1, 2ᕫ 1 2 x x = = 0, 0 প 3 4 x x = = 0, 19 ᕫ 1 2 x x = = 1, 7 ℸ⸔㾷㋏Ў 1 2 0 1 0 7 , 1 0 2 19 æö æ ö ç÷ ç ÷ = = ç÷ ç ÷ ç÷ ç ÷ èø è ø ȟ ȟ 17ˊ䆒 2 213 9 528 æ ö - = ç ÷ è ø - A ˈ∖ϔϾ 4 2 ´ 䰊ⶽ䰉 BˈՓ AB = OˈϨ R(B) =2 㾷˖⬅Ѣ R(A)=2,᠔ҹᮍ AX O= ⱘ㾷ぎ䯈ⱘ⾽гᰃ ˈ᠔ҹⶽ䰉 B ህৃ⬅䆹㾷ぎ䯈Ёӏ ԩϸϾ㒓ᗻ᮴݇ⱘ䞣ᵘ៤DŽ 㗠 1 3 1 1 1 1 1 0 2 213 2 2 8 8 ~ ~ 9 5 2 8 5 11 5 11 0 4 01 22 88 æ ö æ ö - - - ç ÷ ç ÷ æ ö = ç ÷ ç ÷ ç ÷ è ø - ç ÷ -- -- ç ÷ è ø è ø A ᠔ҹᮍ AX O= ⱘ⸔㾷㋏Ў 1 2 1 1 5 11 , 8 0 0 8 æö æ ö - ç÷ ç ÷ = = ç÷ ç ÷ ç÷ ç ÷ èø è ø ȟ ȟ ᬙ᠔∖ⶽ䰉 1 1 5 11 8 0 0 8 æ ö - ç ÷ = ç ÷ ç ÷ è ø B ˊ 18ˊ∖ϔϾ唤㒓ᗻᮍ㒘ˈՓᅗⱘ⸔㾷㋏Ў T T 1 2 ȟ = = (0,1,2,3) , (3,2,1,0) ȟ 㾷˖ᰒ✊᠔∖ᮍ㒘ⱘ䗮㾷Ў
