I cannot solve the following exersice

Question

Prove that H = {(2k,3k) when k in R} is a subgroup of( R^2,+)

Answer 1

What you need to prove is that the subset H of R^2 also forms a group with addition (that is what a subgroup means).

Let's us check if H is a group with addition.

1. Does the sum of any two elements of H belongs to H? Yes ( (2k,3k)+(2l,3k)=(2(k+l),3(k+l)) and k+l is real since both k and l are real)
2. Associativity? clearly satisfied
3. Zero element? Same as for (R^2,+) i.e. (0,0)
4. Inverse belong to the group? Since group operation is addition, the inverse of (2k,3k) is (-2k, -3k) which clearly also belongs to H

All group properties satisfied for (H,+) => H is a subgroup of (R^2,+). HTH.