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㄀ಯ⃵԰Ϯ 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ˈᄬ೼ϸ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