I was wondering if there is a generalization of this theorem from finite numbers to cardinal numbers.

More specifically, let $\displaystyle F$ be a field and $\displaystyle V$, $\displaystyle W$ be vector-spaces over $\displaystyle F$.

Let the dimension of $\displaystyle V/F = \aleph_{\alpha}$.

Let $\displaystyle T:V\mapsto W$ be a linear transformation then is $\displaystyle \mbox{dim}(\mbox{ker} T) + \mbox{dim}(\mbox{Im} T) = \aleph_{\alpha}$?