Let $U,V\subset W$, then the sum $U+V$ is direct and equals $W$ if and only if $U+V=W$ and $U\cap V=\{0\}$. And in this situation say $W$ is the internal direct sum of $U$ and $V$. The definition of sum being direct is $f(u,v)=u+v:U\bigoplus V\implies U+V$ is an isomorphism.
The proof says this is true because $f$ is an isomorphism if and only if $Im(f)=W,Ker(f)=\{0\}$ why?
Here I already know the theorem $\dim_F(U+V)+\dim_F(U\cap V)=\dim_F(U)+\dim_F(V)$ but I don't think it's helpful here. But let me know if you think it is.
Also it says the definition of $U+V$ being direct is equivalent to $U\cap V=\{0\}$ why?
Edit: Okay I think I see the reversed direction. First Isomorphism Theorem tells the thing.