I was wondering if there is a generalization of this theorem from finite numbers to cardinal numbers.
More specifically, let be a field and , be vector-spaces over .
Let the dimension of .
Let be a linear transformation then is ?
I was wondering if there is a generalization of this theorem from finite numbers to cardinal numbers.
More specifically, let be a field and , be vector-spaces over .
Let the dimension of .
Let be a linear transformation then is ?
your question basically is whether or not we have for any subspace of
where here means the cardinality of a basis of a vector space. the answer is yes! let
be a basis for and a basis for let it's easy to see that