It sounds to me like you mean "rank of a transformation" as in the dimension of the transformation's image.
If that is the case, then if you express the transformation by left multiplication with a matrix $A$, then the dimension of the columnspace of $A$ is exactly the rank of $T$. The column vector you are operating on is merely making a linear combinations of the columns of $A$.
There are several terms floating around here that are almost the same.
- rank of a linear transformation (there is no such thing as "the" columnspace or rowspace of a transformation, but the dimension of the image is well-defined.)
- column rank of a matrix (dimension of columnspace of the matrix)
- row rank of a matrix (dimension of the rowspace of a matrix)
I think you will be able to sort it all out from this wiki article.