Question

For each set of three vectors, find a value of d that makes them linearly dependent. If no such value exists, enter DNE. (a) {[5,1,2).[1,2,3].[6 +0,3 +2 2,5+3 d } (b) {(1,1,5),(3,2,5),(4,3,2]} (c) {(1,5,1),(3, 15,4),(4,1,2] }

Answer 1

(a) The { given set of vectors is } = {15,1,2),(1,2,3) (64d, 3+2d, 5t3d} A= =15 n n I 6+d 1 N A LV 31 -1 2 3+2d 2 ] 5+3dJ The given set of vectors is linearly dependent, it det (A)=0 15 I 6+d =0 1 2 3+20 12 3 5+3d| 512 3+2d-: 3+2d| +6+4)|| 21:0 3 stadl 12 5+3d1 12 31 5 (10+6d-9-6d)-1(5+3d-6-4d) ++d) (3-4)=0 5(1)-1(-1-d) +(6+2) (-10=0 stlt.d-6-d=0 020, which is true, and does not depend on de .. The given set of vectors is linearly dependent for any real value of dl

47 (b) The given set of vectors is {v,, , ;} = {[1,1,5][3, 2,5), (4,3,d]} A= {i ų į ] = [ 3 4 15 I do The given set is linearly dependent, it det (A)=0 > 113 31-313 314/s 3120 => 1 (2d-15)-3d-15) +4(5-10)_0 = 2d-15-3d+45+41-5)=0 -d +10=0 d=/10 (C) The given set of vectors is {%, į } = {(1,5,1],[3,15,4), (9,ld]} A={ů į ] = [3 at 4 d] det (A) = 11 IS is 1 =0 14 d 1(15d-4)-3(5d-1) +4(2015)=0 15d-4-15d +3 +20=0 1920 which is incorrect id=LDNE)