I would like to ask if you may know how to prove this claim or any theorem related:

Given 9 points (x,y,z) lie on unit sphere in 3 dimensional space such that any 4 points are not on the same plane. Let vector a = [1, x, y, z]. Form the 16x9 matrix A = [a1⊗a1, a2⊗a2,.. ,a9⊗a9]. Prove rank of A is 9. (where is the tensor product or the Kronecker product)

I used matlab to confirm this claim, but I wonder if there is any concrete proof for that. Thank you in advance.