Give an example of a linear transformation $$\begin{array}{l}T:{P_2}(x) \to {P_3}(x)\\{\mathop{\rm Im}\nolimits} (T) = Sp\{ {x^3} + 1,{x^2},2{x^3} + 2{x^2} + 3\} \end{array}$$
So, $T(v)$ must be linear-dependent on those four vectors.
I tried to express $T(v)$ with four scalars, but couldn't reach much further than that.  
I need a direction.
Thanks!

I think you can set: $$T(1)=x^3+1,~~T(x)=x^2,T(x^2)=2x^3+2x^2+3$$ because we are allowed to do this. In fact, there is an theorem which tells that between two vector space defined on a field $F$, such this unique transformation exists. See Ittay's proof for it.
