Given a bilinear map $B:X\times Y\to F$ where $X,Y$ are vector spaces and given $S\leq X$, why is $\dim S+\dim \operatorname{ann}(S)=\dim Y$ where $\operatorname{ann}(S)$ is the annihilator of $S$ viz. $\operatorname{ann}(S)=\{y\in Y|B(S,y)=0\}$? N.B. $B$ is may or may not be degenerate.
Added in repsonse to Jan's comment: Perhaps a more accurate statement would be $\dim S+\dim \operatorname{ann}(S)\geq\dim Y$? Is this right? How might I prove this?
Thanks.