if $\phi): RxR \(\to$ R, and ker$\phi$={(x,y)$\in$ RxR | x+y=0}; what is RxR/ker$\phi$ isomorphic to. What is a description of that?

Question

if $\phi): RxR \(\to$ R, and ker$\phi$={(x,y)$\in$ RxR | x+y=0};  what is RxR/ker$\phi$ isomorphic to.  What is a description of that?

amistre64 · Answer

almost got the coding right lol ....

amistre64 · Answer

$\phi$: RxR $\to$ R

amistre64 · Answer

i spose my real question is,  what does a factor group describe?

my current thinking is that a factor group is a coset, H, that divvys up the group into equal partitions.  Does H have to be normal in the group?

I think ker$\phi$ is normal to the group ... last night i was thinking maybe the desciption is:

since the kernel of phi is the set of ordered pairs (x,y) such that y=-x, this is a coset of RxR such that it is divided into partitons of parallel lines such that form some (a,b)H is the set of ordered pairs (x,y) such that y-b = -(x-a)

anonymous · Answer

Just curious to which field of mathematics your question belongs to ?

amistre64 · Answer

it was a question on my abstract algebra final

amistre64 · Answer

so im assuming none of the fields :)

anonymous · Answer

Is $\mathbb{R}\times \mathbb{R}$ being considered as a group under addition?

amistre64 · Answer

dunno ... i think the definition it used was:$$\phi(\,(x,y)\,)=x+y$$

amistre64 · Answer

and stated phi was a homomorphism

amistre64 · Answer

addition seems familiar .... might have been addition of vectors in RxR

anonymous · Answer

ok
but in order for it to make sense, then RxR/ker assumes that RxR is something
a group, ring, something

amistre64 · Answer

addition makes the most sense since we were spose to show that:$$\phi(ab)=\phi(a)\phi(b)$$

amistre64 · Answer

then lets go with group,  RxR with operation of addition of vectors

anonymous · Answer

not that i know the answer, but i have a way to think about it, informally at least
things in ker f are $(a,-a)$  so for example $(5,2)$ would be in the same coset as $(3,0)$ because if you are thinking of it as a group under addition, then $(5,2)=(3,0)+(2,-2)$

anonymous · Answer

well i screwed that up, didn't i?

anonymous · Answer

what i meant was the same coset as $(7,0)$

anonymous · Answer

in other words, $(b,a)$ would be in the same coset as $(b+a,0)$

anonymous · Answer

making it isomorphic, if my intuition is correct, to $\mathbb{R}$

anonymous · Answer

i would only bet $6 on this answer, because i am only 60% sure it is right

amistre64 · Answer

thats making sense, since the coset is made of the elements in RxR, then aH or Ha is the operation in RxR

anonymous · Answer

unfortunately i gotta  run to class i will think about it more later
but yes, what you said, except since the operation is (i assume) pointwise addition, it would make more sense to write $a+H$  where of course $a=(x,y)$

anonymous · Answer

another intuitive way to think about it
RxR is a plane
x +  y = 0 is a line
mod out by the line, you get a line

anonymous · Answer

or yet another way
all other cosets are parallel to $x+y=0$ 
they are all the same, just differ by the y intercept
boy am i going to be embarrassed if this is wrong ....

amistre64 · Answer

the parallel line was what i was thinking,  similar to a notion of Z4 forms equivalence classes by considering all the remaniders left when dividing by 4;  this current setup may just as well be a set of equivalence classes of affine lines by considereing their y intercepts.