1. First Isomorphism Theorem

Let S,T be subspaces of the vector spaces V ,U (respectively), and let

$\displaystyle T : V × U -> (V/S)× (U/T)$ be the linear map defined by $\displaystyle T(x, y) = (x +S, y + T).$ Find the kernel of T and prove that

$\displaystyle (V× U)/(S× T) (~ =) (V/S) × (W/T).$

Plx help...

2. First, do you mean to define $\displaystyle T(x, y) = (x +S, y + T)$?

I think that your first step is to determine the kernel of $\displaystyle T$. If $\displaystyle T(x,y)=(0 +S, 0 +T)$, what can be said about $\displaystyle x$ and $\displaystyle y$?

3. Originally Posted by roninpro
First, do you mean to define $\displaystyle T(x, y) = (x +S, y + T)$?

I think that your first step is to determine the kernel of $\displaystyle T$. If $\displaystyle T(x,y)=(0 +S, 0 +T)$, what can be said about $\displaystyle x$ and $\displaystyle y$?
is it Ker T= T(0,0)=(0+S, 0+T) =(S,T)
therefor Ker T = (S,T)?

but I still don't know how to prove the following....

4. Originally Posted by mathbeginner
is it Ker T= T(0,0)=(0+S, 0+T) =(S,T)
therefor Ker T = (S,T)?

but I still don't know how to prove the following....
You have that $\displaystyle F:V\boxplus U\to (V/S)\boxplus (U/T)x,y)\mapsto \left(x+S,y+T)$ (I use $\displaystyle \boxplus$ to differentiate between external and internal direct sum), this is evidently a surjective linear homomorphism. Then, we are trying to look for what values $\displaystyle (x,y)$ is $\displaystyle T(x,y)=...$...equals what? We're trying to look for the origin in the vector spaces $\displaystyle V/S$ and $\displaystyle U/T$ but these are precisely $\displaystyle S$ and $\displaystyle T$ respectively. So, for what $\displaystyle x$ is $\displaystyle x+S=S$ and for what $\displaystyle y$ is $\displaystyle y+T=T$? It is fairly clear that these statements are true if and only if $\displaystyle x\in S$and $\displaystyle y\in T$. Thus, a quick proof shows that $\displaystyle \ker F=S\boxplus T$ and so by the F.I.T. $\displaystyle \left(V\boxplus U\right)/\left(S\boxplus T\right)\cong \left(V/S\right)\boxplus\left(U/T\right)$

5. I don't reli get why Ker F = S X T and how to prove that???

6. Originally Posted by mathbeginner
I don't reli get why Ker F = S X T and how to prove that???
Come on friend. Review your definitions. You're trying to show that $\displaystyle \ker F=S\boxplus T$ which is equivalent to showing $\displaystyle F(x,y)=(S,T)\Leftrightarrow (x,y)\in S\boxplus T$. I give the solution below "hidden". Try your damnedest, and only look at it as a last result.
[spoil]Clearly if $\displaystyle (x,y)\in S\boxplus T$ then $\displaystyle F(x,y)=(x+S,y+T)=(S,T)$ since $\displaystyle x+S=\left\{x+s:s\in S\right\}=S$ since $\displaystyle S\leqslant V$ and similarly $\displaystyle y+T=T$. Conversely, suppose that $latex F(x,y)=(S,T)[/tex] then$\displaystyle x+S=S$. But, note that$\displaystyle x\in x+S$since$\displaystyle 0\in S$and so$\displaystyle x\in S$. Similarly$\displaystyle y+T=T\implies y\in T$and so$\displaystyle F(x,y)=(S,T)\implies (x,y)\in (S,T)$. Combining these we get$\displaystyle F(x,y)=(S,T)\Leftrightarrow (x,y)\in S\boxplus T\$[/spoil]

EDIT: I can't figure out how to hide it...so work on the honor code haha

7. Originally Posted by Drexel28
EDIT: I can't figure out how to hide it...so work on the honor code haha
[Spoiler] [/Spoiler] You have been gone for so long that you forgot the tags!

8. Originally Posted by TheCoffeeMachine
[Spoiler] [/Spoiler] You have been gone for so long that you forgot the tags!
Haha, I guess so!