Let A B and C Linearly Dependent 1 Determine whether

Let A = , B = , and C = . Linearly Dependent : 1. Determine whether or not the three vectors listed above are linearly independent or linearly dependent. If they are linearly dependent, determine a non-trivial linear relation. Otherwise, if the vectors are linearly independent, enter 0s for the coefficients, since that relationship always holds. A + B+ C = 0.

Solution

The three vectors are said to be linearly independent, if one vector can\'t be expressed as the sum of the coefficients of other two vectors or

aV1 + bV2 + cV3 = 0 [ only in the case, when all coefficients a,b,c are 0]

substituting the vector V1, V2 and V3 we get

a [ 2 3 -6] + b [ 12 3 -26] + c [2 0 -4] = [0 0 0]

2a + 12b + 2c = 0 ----- (i)

3a + 3b = 0 ---- (ii)

-6a - 26b - 4c = 0 ---- (iii)

a = -b from (ii)

c = --5b = 5a --- (iii)

so the values which satisfy them can be a=1, b=-1 and c=5

putting back the values we see

1 [2 3 -6] + -1 [12 3 -26] + 5 [2 0 -4]

=> [(2-12+10) (3-3) (-6+26-20)]

=> [0 0 0]

Hence the vectors are linearly dependent with the coefficients are (1,-1,5)