# Range of the set $\theta=\{((x,y),(3y,2x,x+y)):x,y\in\mathbb R\}$ a

So the question posed in the book I'm working through (Book of Proof by Richard Hammack) is as follows:

Is the set $\theta=\{((x,y),(3y,2x,x+y)):x,y\in\mathbb R\}$ a function? If so, then what is its domain, codomain and range?

Domain: $\{(x,y):x,y\in\mathbb{R}\}$
Codomain: $\{(x,y,z):x,y,z\in\mathbb{R}\}$
Hint: the mapping $(x,y) \mapsto (3y,2x,x+y)$ is linear, thus the range of $\theta$ is a subspace of $\mathbb{R}^3$. By finding the matrix associated to $\theta$ you can find a basis of this subspace.