Elementary Linear Algebra Linearly independent?

Yes. You can use Determinant method to find all the values of x.

Steps to be taken:

1)Write all the vectors in matrix form and take its determinant.

2) Set this determinant value to 0 and find x. It will give you all the values of x for which vectors will be linearly dependent.

3)Then all the other values of x except the values you obtained in Step 2 will be your required answer.

They are linearly independent if a(1,-1,0)+b(0,2,3)≠(x,1,0) for all real a and b.

Here -1(1,-1,0)+0(0,2,3)=(x,1,0) when x=-1, so one value of x causes a failure in linear independence. You can show this is the only case.

These vectors are linearly independent when x≠-1

[1 -1 0]
[ x 1 0]
[0 2 3]

The determinant is:
1( 3-0) + 1( 3x-0) = 3 + 3x
3+3x=0
3x=-3
x=-1

If x ≠ -1, the vectors are linearly independent