I apologize if this question is "too elementary" - I am not an algebraic geometer. Are the following statements true?

Let $k\subset K$ an extension of algebraically closed fields of characteristic 0, and $f_{1},\ldots, f_{m}\in k[x_{0},\ldots, x_{n}]$ homogeneous polynomials. Consider the subschemes $S_{k}\subset\mathbb{P}^n_{k}$ and $S_{K}\subset\mathbb{P}^n_{K}$ given by $f_{1}=\cdots=f_{n}=0$ (to define $S_{K}$ we just use $k[x_{0},\ldots, x_{n}]\subset K[x_{0},\ldots, x_{n}]$).

Then (1) $\dim S_{k} = \dim S_{K}$, (2) if $\dim S_{k} = \dim S_{K}=0$ then $S(k)=S(K)$.