Componentwise Addition

jujab · #1 $Old$ November 3rd, 2009, 08:19 PM

In R + R under componentwise addition, let H be the subgroup of all points lying on a line through the origin. Show that any left coset of H is just a line parallel to H. Draw a sketch illustrating a nonidentity coset.

Any help is appreciated greatly!

tonio · #2 $Old$ November 3rd, 2009, 09:09 PM

Quote:

Originally Posted by jujab $View Post$

In R + R under componentwise addition, let H be the subgroup of all points lying on a line through the origin. Show that any left coset of H is just a line parallel to H. Draw a sketch illustrating a nonidentity coset.

Any help is appreciated greatly!

A line through the origin looks like $\{(x,mx)\}$ , for some real m, and a left coset of this looks like $(a,b)+\{(x,mx)\}=\{(a+x,mx+b)\}$ , which is just the line $y=m(x+a)+(b-ma)$ , which has the same slope as the first one and is thus parallel to it.

Tonio

jujab · #3 $Old$ November 3rd, 2009, 09:22 PM

Thanks Tonio!