Find isomorphic vector spaces $V$ and $W$ with $V = S \bigoplus B$ and $W = S \bigoplus D$ but $B \ncong D$.

Question

I'm having some trouble with this. It seems very difficult to construct the spaces in such a way that $ V = S \bigoplus B$ and $W = S \bigoplus D$ with the conditions that $ B \ncong D$.

Any help would be great. Thanks.

Answer 1

Hint: A vector space over a given field is determined up to isomorphism class by the cardinality of its dimension. Look below for a second hint:

In cardinal arithmetic, $\aleph_0 + \aleph_0 = \aleph_0$.

And the solution is:

Take $S$ to be of countably infinite dimension, $B$ to be finite dimensional, and $D$ to be of countably infinite dimension.