Hi,

This is a part of an exercise I got on the final exam I failed. I couldn't answer well this question : Let $\displaystyle V=\mathbb {R}^4$ and $\displaystyle W=P_3$ the space of polynomials of degree $\displaystyle \leq 3$.

a)Show that $\displaystyle V$ is isomorph to $\displaystyle W$. (Done)

b)Does there exist 2 distinct isomorphisms? If so, give them. (Done. The answer was yes if it might help for the next question)

c)Tell whether there exist a linear transformation $\displaystyle T:V\to W$ such that $\displaystyle T(1,0,0,0)=x$, $\displaystyle T(0,0,0,2)=x$, $\displaystyle T(1,1,1,1)=x$ and $\displaystyle T(1,0,0,-2)=x$.

My attempt :It doesn't exist because the vectors $\displaystyle (1,0,0,0)$, $\displaystyle (0,0,0,2)$, $\displaystyle (1,1,1,1)$ and $\displaystyle (1,0,0,-2)$ are linear independent. And so $\displaystyle T(1,0,0,0)$, $\displaystyle T(0,0,0,2)$, $\displaystyle T(1,1,1,1)$ and $\displaystyle T(1,0,0,-2)$ should... I stopped right there. I'm not sure of how to answer the question.