Consider the space of all homogeneous degree $d$ polynomials in three variables $[X,Y,Z]$, i.e. $$ f[X,Y,Z] = f_{d00} X^d + f_{d10} X^{d-1}Y + \ldots .$$ This can be thought of as a section of the line bundle

$$ \gamma_{\mathbb{P}^2} ^{* d} \rightarrow \mathbb{P}^2 $$

Upto scaling this is the projective space $\mathbb{P}^{\delta_{d}}$, where
$\delta_{d} = \frac{d(d+3)}{2}.$

Note that ``passing through a point '' imposes a linear condition on the
coefficients $f_{ijk}$.
Fix any number $k \leq \delta_{d}$.
Is there
an abstract way to show that there exist $k$ points
$[X_1, Y_1, Z_1]\ldots, [X_k, Y_k, Z_k]$ such that passing through those
$k$ points imposes linearly independent conditions on the coefficients?
More precisely, the conditions will not be linearly independent implies
that a certain determinant involving the $[X_i,Y_i, Z_i]$ is zero.
How do I know that this determinant is not $\textit{always}$ zero?

One can explicitly write out the matrix whose determinant should
not be zero and try to produce some explicit choice of points
for which the determinant is not zero. But is there a way to avoid
doing that? Does the Bertini theorem (or some modification of that)
imply that there exists $k$ points so that passing through $k$
points imposes linearly independent conditions?
In any case what is the simplest way to prove this seemingly
obvious statement?
Everything is over the
complex numbers.