Determine a non trivial linear relation

Question

Let A= [0,3,0,5] B=[2,-3,-1,-1] C=[2,21,8,33] D=[-2,6,4,4]

 

 they are linearly dependent, determine a non-trivial linear relation 

Answer 1

If a set of vectors are linearly dependent, then aA + bB + cC + dD = 0 with a, b, c, and d not all zeros.

 

aA + bB + cC + dD = 0

 

a[0,3,0,5] + b[2,-3,-1,-1] + c[2,21,8,33] + d[-2,6,4,4] = [0,0,0,0]

 

 

Equating components,

 

2b + 2c -2d = 0

 

3a -3b + 21c + 6d = 0

 

-b + 8c + 4d = 0

 

5a - b + 33c + 4d = 0

 

 

 

a = 5, b = 4, c = -1, and d = 3 would be a possible set to show linear dependence.