The question came up while reading a bit more into the Hilbert-Zariski theorem I asked about the other week.
Suppose $V$ is an algebraic variety over arbitrary field $k$. (For this situation, I'll take the definition $\dim\ V=\deg_k(k(x))$, where $(x)\in V$ is a generic point, and by $\deg$ I mean the transcendence degree.) As usual, $V(f_1,\dots,f_s)$ is the set of zeroes of the homogeneous forms $f_1,\dots,f_s)$ in the affine space.
Let $\dim\ V=d$. Suppose I take $d$ generic linear forms $f_1=\sum_{i=1}^n a_{1i}X_i,\dots,f_d=\sum_{i=1}^n a_{di}X_i$, so that the $a_{ji}$ are algebraically independent. In this scenario, is it true that $ V(f_1,\dots,f_d)\cap V=\{0\}? $
If so, why is there only the trivial zero? Many thanks.