Suppose $E$ is a vector space over a field of characteristic $0$. Let $E_1, F_1$ be subspaces of finite codimension and let $E_2, F_2$ be their respective complements, i.e., $E = E_1 \oplus E_2 = F_1 \oplus F_2$.$\DeclareMathOperator{\codim}{codim}$
I know that $\dim E_2 = \codim E_1$ and $\dim F_2 = \codim F_1$ because $E_2 \cong E/E_1$ and $F_2 \cong E/F_1$.
But I don't know how to prove that $\codim (E_1 \cap F_1) \le \dim E_2 + \dim F_2$. I saw a proof that $\codim (E_1 \cap F_1)$ is finite but it used some fancy isomorphism theorem, so I think a more bare hands approach would be helpful. Thanks.
Best Answer
Maybe there is some way for avoiding the homomorphism theorems, but they're so handy and powerful that it is better trying to understand them.
Consider the linear map $$ f\colon E\to E/E_1\oplus E/F_1,\qquad f(v)=(v+E_1,v+F_1) $$ The kernel of this map is $E_1\cap F_1$, so $f$ induces an injective linear map $$ \tilde{f}\colon E/(E_1\cap F_1)\to E/E_1\oplus E/F_1 $$ In particular, the domain is finite dimensional and $$ \dim E/(E_1\cap F_1)\le\dim(E/E_1\oplus E/F_1)= \dim(E/E_1)+\dim(E/F_1) $$ which is the same as saying that $$\DeclareMathOperator{\codim}{codim} \codim(E_1\cap F_1)\le\codim E_1+\codim F_1 $$