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Vectors Linear Independent 
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#1
Aug2509, 11:21 AM

P: 12

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 


#2
Aug2509, 11:51 AM

P: 365

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. 


#3
Aug2509, 12:02 PM

P: 12

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 


#4
Aug2509, 12:23 PM

P: 365

Vectors Linear Independent
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. 


#5
Aug2509, 12:36 PM

P: 12

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 + (q1) = 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 


#6
Aug2509, 12:56 PM

P: 365

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=q1 p=32q q any number in R, you chose it q=1 Regards. 


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