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(Crying)

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- Oct 28th 2010, 08:07 AMmathbeginnerDirect sum and Isomorphisms
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(Crying) - Oct 28th 2010, 07:58 PMtonio

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 - Oct 28th 2010, 08:22 PMDrexel28
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.

- Oct 28th 2010, 09:50 PMmanygrams
Would you happen to be a uOttawa student? I have that same question on an assignment of mine