Let $a=(1,0,3)$ $b=(0,2,0)$ $c^*=(1,2,3)$ be elements of $\mathbb R^3$ and let $x=(1,3)$ $y=(0,2)$ $z=(0,1)$ $z^*=(1,5)$ be elements of $\mathbb R^2$. Do there exist linear maps which satisfy
i) $f:a\mapsto x,b\mapsto y,c^*\mapsto z$
ii) $f:a\mapsto x,b\mapsto y,c^*\mapsto z^*$
In each case state if the linear map is unique giving reasons
for ii) the map $(a,b+c)$ work but i'm not sure if it's unique or not and I can't find a map for i) so I'm not sure if there is one or not.