Existence with values on dependent vectors

Determine whether there exists a linear map $f:{\mathbb{R}}^2\to {\mathbb{R}}^3$ such that

$$f(1,0)=(1,1,0),f(1,1)=(1,0,1),f(2,1)=(2,1,1);$$

if so, determine whether it is unique and whether it is injective.