Let $V$ be a direct sum of two subspaces $U,W$. Prove that $U\times W$ is isomorphic to $V$?
My problem lies in the fact the book has not provided any formal definition or theorem about $U\times W$, I mean about the product of Vector Spaces, on this case subspaces. As far as I can see the $ker(T)=0$, because $0\times 0$ is preserved but I cannot talk about dimensions.
Could anyone provide me a definition of $U\times W$?
How can I prove $U\times W$ is isomorphic to V?
Thanks in advance!