Prove that if V = M (Direct sum) N, then
V/M (Isomorphisms.) N. Hint: Restrict the quotient mapping V /M to N and the kernel and image of the restricted mapping.
plx help
As any $\displaystyle v\in V$ has a unique expression as $\displaystyle v=m_v+n_v\,,\,\,m_v\in M\,,\,n_v\in N$ , define $\displaystyle f:V\to N$ by $\displaystyle f(v=m_v+n_v):= n_v$
Uniqueness of expression gives you that f is well defined, and now just prove that f is an isomorphism of vec. spaces.
Tonio
Pd. Of course, f is NOT an isomorphism! You need to find its kernel and use the fist isomorphism theorem
Assuming you're working with F.D. vec. spaces if you only need to show that they are isomorphic and you need not exhibit the isomorphism why not note that if $\displaystyle \{y_1,\cdots,y_n\}$ is a basis for $\displaystyle \mathcal{N}$ then $\displaystyle y_1+\mathcal{M},\cdots,y_n+\mathcal{M}$ is a basis for $\displaystyle \mathcal{V}/\mathcal{M}$. In particular, $\displaystyle \dim \mathcal{V}/\mathcal{M}=n=\dim\mathcal{N}$ and so they are evidently isomorphic.