1. The problem statement, all variables and given/known data Considering the following vectors R[tex]^{4}[/tex]: v1 = (1,2,0,2) v2 = (2,3,1,4) v3 = (0,1,-1,0) Determine if these vectors are linearly independent. Let S be the linear span of the three vectors. Define a basis and the dimensions of S. Express the vector v=(3,5,1,6) as a linear combination of the three vectors. Can this be achieved in a unique way? Justify your answer? 2. Relevant equations I tried to put it into matrix form and reduce via row echolon but I'm not if this is the correct or proper way 3. The attempt at a solution [ 1 2 0 2 2 3 1 4 0 1 -1 0 3 5 1 6] [ 1 2 0 2 0 -1 1 0 0 1 -1 0 0 0 0 0 ] x +2y = 2 y - z = 0 -y + 2 = 0 therefore y=z making it linearly independent
You need to prove that p=q=r=0, for v1,v2,v3 to be linear independent: [tex]pv_1 + qv_2 +rv_3=0[/tex] [tex]p(1,2,0,2)+q(2,3,1,4)+r(0,1,-1,0)=0[/tex] You should express the vector v in same manner as linear combination of v1,v2,v3: i.e pv1+qv2+rv3=v p,q,r are random scalars. Regards.
so with that being said which of the two do I follow from below to work out the answer? a) 1p + 2q = 0 2p +3q +r = 0 q - r = 0 2p + 4q = 0 b) 1p + 2q = 3 2p +3q +r = 5 q - r = 1 2p + 4q = 6 and if I follow b I'm I right to think that p = 1 q =2 and r = 0
Ok, your task have two parts, a) to check the linear independence of the vectors v1,v2 and v3 b)to find out if the vector v can be represented as linear combination of the vectors v1,v2 and v3. So you need to solve both a) and b). Regards.
a) 1p + 2q = 0 (1) 2p +3q +r = 0 (2) q - r = 0 (3) 2p + 4q = 0 (4) (3) q = r (1) p = -2q put (3)and(1) into (2) 2(-2q) + 3(q) +q = -4q +3q + q = 0 p=-2 q = 1 r = 1 vectors are dependent b) 1p + 2q = 3 (1) 2p +3q +r = 5 (2) q - r = 1 (3) 2p + 4q = 6 (4) (3) q - 1 = r (3) into (1) 2p + 3q + (q-1) = 5 ; 2p +4q = 6 (same as 4) (4) can be divide by 2 to equal (1) answer therefore is p = 1 q = 1 r = 0 so it that then correct? Thank you by the way your really helpful
I am glad that I helped you. Just a little correction: a) r=q p=-2q q any number in R, you chose q=1 The vectors are linear dependent b) r=q-1 p=3-2q q any number in R, you chose it q=1 Regards.