Advanced Algebraic Curves problem

**[Question]**

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]

respectively.

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.