Show that the dimention of the intersection of projective linear sub-spaces of dimentions $d_1$ and $d_2$ of $\mathbb{P}^n$ is bigger than $d_1+d_2-n$

Solution 1:

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 $d_1+d_2-n$.