Determine a non trivial linear relation

Question

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

 

We know that they are linearly dependent

 

1) Determine a non-trivial linear relation (a non trivial linear relation is three numbers which are not 0)

?A+?B+?C+?D=0

Answer 1

As I mentioned in my solution to one of your other problems, you can solve this by setting up a system of equations based on each unknown coefficient.

 

If w A + x B + y C + z D = 0,

 

then you can write 4 equations, starting with

 

w*0 + x*2 + y*2 + z*-2 = 0

 

and solve the system for those 4 variables using your favorite method. The algebra for this is tedious to do by hand; WolframAlpha suggests a solution starting with w=5.