wow hard vector question?

Question

<0,0,0,1><0,0,2,1><0,3,2,1><0,1,-1,1> I know these vector are linearly dependent but I cannot find the solution to
w=0 2z+w=0 3y+2z+w=0 y-w+z=0 other than w=x=y=z=0.

Answer 1

That's because those aren't the right equations.

You need to solve the equation
x<0,0,0,1> + y<0,0,2,1> + z<0,3,2,1> + w<0,1,-1,1> = <0,0,0,0>
i.e.
0 = 0 (first coordinate)
3z + w = 0 (second coordinate)
2y + 2z - w = 0 (third coordinate)
x + y + z + w = 0 (fourth coordinate)

So we get w = -3z, 2y = w - 2z = -5z so y = -5z/2, and x = -y-z-w = 5z/2 - z + 3z = 9z/2.
e.g. take z = 2, then we get x = 9, y = -5, z = 2, w = -6 to give

9<0,0,0,1> - 5<0,0,2,1> + 2<0,3,2,1> - 6<0,1,-1,1> = <0,0,0,0>

You can verify that this is true.

Answer 2

make a free variable.