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Math Help - Componentwise Addition

  1. #1
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    Componentwise Addition

    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!
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  2. #2
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    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
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    Thanks Tonio!
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