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 $\displaystyle F$ 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 $\displaystyle \mbox{dim}(\mbox{ker} T) + \mbox{dim}(\mbox{Im} T) = \aleph_{\alpha}$?

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 $\displaystyle F$ 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 $\displaystyle \mbox{dim}(\mbox{ker} T) + \mbox{dim}(\mbox{Im} T) = \aleph_{\alpha}$?
your question basically is whether or not we have $\displaystyle \dim V = \dim K + \dim(V/K),$ for any subspace $\displaystyle K$ of $\displaystyle V,$

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

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

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