Proposition:
Let $L$ and $M$ linear projective subspaces of $\mathbb{P}^n$ of dimention $d_1$ and $d_2$ respectively. Prove that $\operatorname{dim}(L\cap M)\geq d_1+d_2-n$ (we consider the dimention of empty set is $-1$). When occur the equalty?
Remark: Let $L\subset \mathbb{P}^n$ a projective sub-variety defined by $L=(f_1,f_2,\ldots, f_s)$ where the $f_i$ are homogenous polynomials of degree $1$ and linearly independent.Such $L$ is called linear projective subspace.
Attempt:
I use the fact if $L$ is a linear projective space then $L\simeq \mathbb{P}^{n-s}$, in particular $\operatorname{dim}L=n-s$.Let us consider $L=V(f_1,f_2,\ldots,f_{d_1})$ and $M=V(g_1,g_2,\ldots,g_{d_2})$ two linear projective varietys where $f_i,g_j$ are homogenous polynomials of degree $1$ and linearly independent.
Now $$L\cap M= V(f_1,f_2,\ldots,f_{d_1})\cap V(g_1,g_2,\ldots,g_{d_2})=V((f_1,f_2,\ldots,f_{d_1})+(g_1,g_2,\ldots, g_{d_2}))$$ and hence $$L\cap M=V(f_1,f_2,\ldots, f_{d_1},g_1,g_2,\ldots, g_{d_2})$$ And by the lemma $\operatorname{dim}(L\cap M)=n-d_1-d_2$ as well.
But this doesn´t match with the result.
I don´t figure out where I get a fault and also how I should procede to prove that inequality. Any suggestion was useful. Thanks in advice.
First, some minor accounting business: if $L$ is of dimension $d_1$, it's of codimension $n-d_1$, and therefore $I(L)=(f_1,\cdots,f_{n-d_1})$, and similarly $I(M)=(g_1,\cdots,g_{n-d_2})$.
You've successfully shown that $I(L\cap M)=(f_1,\cdots,f_{n-d_1},g_1,\cdots,g_{n-d_2})$, but this is not necessarily a linearly independent generating set of polynomials: you might have to throw some away in order to achieve linear independence. So $I(L\cap M)$ is generated by at most $(n-d_1)+(n-d_2)$ linearly independent linear polynomials, therefore it is of codimension at most $2n-d_1-d_2$, or of dimension at least $d_1+d_2-n$.