Dimension of the system of hypersurfaces of degree $d$ that passes through a point with multiplicity $m$. A hypersurface of degree $d$ of $\mathbb{P}^{n}$ comes from a expression like
$$
F=\displaystyle\sum_{i_{0}+\cdots+i_{n}=d}U_{i_{0},\ldots,i_{n}}X_{0}^{i_{0}}\cdots X_{n}^{i_{n}}.
$$
The hypersurfaces of degree $d$ that passes through a point $p=(p_{0}:...:p_{n})\in\mathbb{P}^{n}$ are those whose coefficients satisfy the equation
$$
\displaystyle\sum_{i_{0}+\cdots+i_{n}=d}U_{i_{0},\ldots,i_{n}}p_{0}^{i_{0}}\cdots p_{n}^{i_{n}}=0,
 $$
which is a hyperplane of $\mathcal{L}^{n}_{d}$, the projective space of hypersurfaces of degree $d$ in $\mathbb{P}^{n}$.
Now, the hypersurfaces that passes through $p$ with multiplicity at least $m$ are those who satisfy the $\binom{m+n-1}{n}$ linear conditions given by the vanishing of the coefficients of the Taylor series expansion of the non-homogeneous polynomial asociated to $F$. My question is the next:
These linear conditions are supposed to be independent, but how do we know?
 A: Suppose we are in characteristic zero. The conditions are just that certain coefficients $U_{i_{0},\ldots,i_{n}}$ be zero, hence these conditions on the  coordinates in the parameter space are linearly independent.     
More explicitly, suppose that $p=(1:0:0:\dots:0)$.
If we write $F$ as  $$F(X_0,\dots,X_n)=F_0(X_1,\dots,X_n)X_0^d+F_1(X_1,\dots,X_n)Z^{d-1}+\dots+F_d(X_1,\dots,X_n)X_0^d$$ (with $F_i$ homogeneous of degree $i$) the condition that the hypersurface $V(F)$ have multiplicity at least $m$ at $p$ is that $F_0=F_1=F_2=\dots=F_{m-1}=0$.
The vanishing of those homogeneous polynomials means that their coefficients are zero and since these coefficients are coefficients of $F$ the conditions are linearly independent coefficients in the coordinates of the projective space $\mathbb P^N$  of dimension $N= \binom {n+d}{n}-1$  parametrizing the hypersurfaces of degree $d$ of $\mathbb P^n$.
Hence there  are $\binom {n+m-1}{n}$ linearly independent conditions and finally the hypersurfaces of degree $d$ having a singularity of order at least $m$ at $p$ form a projective space of dimension $N-1-\binom {n+m-1}{n}$
