I am studying hilbert polynomial from this note: http://www-personal.umich.edu/~stevmatt/hilbert_polynomials.pdf
In particular, example 1:
Let $X=\{p_1,p_2,p_3\}\subset \mathbb{P}^2_k$ be 3 distinct points, then $$h_X(1)=\dim_k(k[X_0,X_1,X_2]/I(X))_1= \dim_k(k[X_0,X_1,X_2]_1)-\dim(I(X)_1).$$
The ideal $I(X)_1$ consists of all homogeneous linear polynomials vanishing at $p_1,p_2,p_3$,
then I don't understand why we can conclude:
$\dim_k(I(X)_1)=1$ when $p_1,p_2,p_3$ are collinear and $0$ otherwise.
How can we see this?