Let p1, p2 and p3 be 3 distinct points in PC2( Projective space, ie
(z0,z1,z2) belong to PC2) Find the dimension of the linear system of
cubics containing these 3 points.

I have solved it for the non collinear case, by taking a projective
transformation of the 3 points to [1,0,0],[0,1,0] and [0,0,1]
And substituting those values into the equation of a cubic, to get
that there are 6 coefficients remaining, therefore the dimension of
the linear system is 6 by definition.

But I am stuck on the collinear case (or maybe it can be shown generally?), thanks in advance for any help
that can be given.