# Direct sum and Isomorphisms

• October 28th 2010, 08:07 AM
mathbeginner
Direct 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)
• October 28th 2010, 07:58 PM
tonio
Quote:

Originally Posted by mathbeginner
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)

As any $v\in V$ has a unique expression as $v=m_v+n_v\,,\,\,m_v\in M\,,\,n_v\in N$ , define $f:V\to N$ by $f(v=m_v+n_v):= n_v$

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
• October 28th 2010, 08:22 PM
Drexel28
Quote:

Originally Posted by mathbeginner
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)

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 $\{y_1,\cdots,y_n\}$ is a basis for $\mathcal{N}$ then $y_1+\mathcal{M},\cdots,y_n+\mathcal{M}$ is a basis for $\mathcal{V}/\mathcal{M}$. In particular, $\dim \mathcal{V}/\mathcal{M}=n=\dim\mathcal{N}$ and so they are evidently isomorphic.
• October 28th 2010, 09:50 PM
manygrams
Would you happen to be a uOttawa student? I have that same question on an assignment of mine