Math Help - Generalized Rank Nullity Theorem

1. Generalized Rank Nullity Theorem

I was wondering if there is a generalization of this theorem from finite numbers to cardinal numbers.
More specifically, let $F$ be a field and $V$, $W$ be vector-spaces over $F$.
Let the dimension of $V/F = \aleph_{\alpha}$.
Let $T:V\mapsto W$ be a linear transformation then is $\mbox{dim}(\mbox{ker} T) + \mbox{dim}(\mbox{Im} T) = \aleph_{\alpha}$?

2. Originally Posted by ThePerfectHacker
I was wondering if there is a generalization of this theorem from finite numbers to cardinal numbers.
More specifically, let $F$ be a field and $V$, $W$ be vector-spaces over $F$.
Let the dimension of $V/F = \aleph_{\alpha}$.
Let $T:V\mapsto W$ be a linear transformation then is $\mbox{dim}(\mbox{ker} T) + \mbox{dim}(\mbox{Im} T) = \aleph_{\alpha}$?
your question basically is whether or not we have $\dim V = \dim K + \dim(V/K),$ for any subspace $K$ of $V,$

where $\dim$ here means the cardinality of a basis of a vector space. the answer is yes! let $A=\{v_{i}: \ i \in I \}$

be a basis for $K$ and $C=\{w_j + K: \ j \in J \}$ a basis for $V/K.$ let $B=\{w_j: \ j \in J\}.$ it's easy to see that

$|B|=|C|, \ A \cap B = \emptyset,$ and $A \cup B$ is a basis for $V.$ Q.E.D.