is it a vector space or not?

Question

check if this is a vector space or not ?

1- let $v=R=\{(x,y):x,y \in \mathbb{R} \}$ check if

$(V,+,.)$ where $(x,y)+(z,w)=(x,y)$ and $k.(x,y)=(k.x,k.y)$ is a vector space or not

2- let $v=R=\{(x,y):x,y \in \mathbb{R}\}$check if $(v,+,.)$ where $(x,y)+(z,w)=(x+z,y+w)$ $k(x,y) = (kx,ky)$

Answer 1

(1) it is not a vector space because $(V,+)$ is not a group (no identity element).

(2) It is a vector space, try to verify it with your definition of vector space (it is easy tell me if you encounter a difficulty).