Are the vectors 2 3 1 3 1 2 2 0 2 2 3 2 and 2 3 1 2 linearly

Are the vectors [2 3 -1 -3], [-1 2 -2 0], [2 2 -3 -2] and [-2 -3 -1 2] linearly independent? If they are linearly dependent, find scalars that are not all zero such that the equation below is true. If they are linearly independent, find the only scalars that will make the equation below true. [2 3 -1 -3] + [-1 2 -2 0] + [2 2 -3 -2] + [-2 -3 -1 2] = [0 0 0 0].

Solution

a [ ] + b [ ] +c [ ] + d[ ] = [0]

assuming lineraly dependent then equations are

2a- b +2c -2d =0------------------1

3a+2b+2c-3d =0-------------------2

-a-2b-3c-d =0-----------------------3

-3a-2c+2d = 0-----------------4

from equation 4) 2c -2d = -3a ------------ 5

replacing that in eqaution 1 ) 2a -b -3a =0

a+b =0 -------- 6

Using equation 5) in equation 2)

2d-2c +2b +2c -3d =0

d = 2b --------------7

using eqation 6,7 in equation 3

-a -2b -3c-d = 0

-(a+b) -b -3c -d =0

d/2 +3c +d =0

3d/2 = -3c

d =-2c ----------------8

in eqation 5 ) 2c -2d = -3a

-3d =-3a

d =a = -2c = 2b

so only possible solution is a=b=c=d = 0

so they are linear independent