If $U_{1}\oplus...\oplus U_{m}=U_{1}+...+U_{m}$ implies that $U_{1}\cap...\cap U_{m}=\{0\}$, how is it that we can say that dim $(U_{1}+...+U_{m})=$ dim $U_{1}+...+$ dim $U_{m}$?
Wouldn't this go against the the true statement dim $(U_{1}+...+U_{m})=$ dim $U_{1}+...+$ dim $U_{m} -$ dim $U_{1}\cap...\cap U_{m}$?
Unless dim $\{0\}$ is zero?
Is the dimension of a set containing only the zero vector equal to zero? I don't see that in my book.