if they are linearly dependent ind a i if 5 Determine whethe

if they are linearly dependent, ind a i if 5) Determine whether the given set of vectors are linearly independent. relation among them. - (1, 1,0)\", t\', _ (0, 1 ,:)r, v, _ (1,0,1)

Solution

If the vectors are linearly independent then we can have the constants a,b,c satisfying the relation such that

av1 + bv2 + cv3 = 0 (where a,b and c all must be zero to satisfy the linear independence)

substituting the values we get

a<1,1,0> + b<0,1,1> + c<1,0,1> = 0

First equation => (a+c) = 0

Second equation => (a+b) = 0

Third equation => (b+c) = 0

adding all the equations we get

2(a+b+c) = 0 => (a+b+c) = 0

This means that the given set of vectors will only be equal to zero in the case when all a,b and c are zero

hence the set of vectors are linearly independent