Isomorphism between subspaces Let $G,H$ be subspaces of a vector space $I$ such that $G \cap H = \{0\}$. Prove that $G + H \cong G\times H$.
I've defined $\phi: G + H \to G\times H$ as $\phi(g + h) = (g,h)$. Showing bijectivity as:


*

*$\phi(g + h) = \phi(j + k) \implies (g,h) = (j,k) \implies g = j, h = k$, so $g + h = j + k$.

*For all $g,h \in G + H$, we have $\phi(g + h) = (g,h)$ so $\phi$ is onto. 
I'm trapped in showing $\phi(hg) = \phi(h)\phi(g)$. 
Is my $\phi$ bad?

You have a short exact sequence:
\begin{alignat*}4
0\longrightarrow G\cap H && \longrightarrow G\oplus H &\longrightarrow G+H \longrightarrow 0 \\
x&&\longmapsto (x,x)\\
&&(x,y)&\longmapsto x-y
\end{alignat*}
As $G\cap H=\{0\}$ by hypothesis, the map $\; G\oplus H \longrightarrow G+H $ is an isomorphism.
