Define a linear transformation $\displaystyle T: P_2 \rightarrow R^2$ by $\displaystyle T(p) = \left[ \begin{array}{cc}P(0)\\P(0) \end{array} \right]$. Find polynomials $\displaystyle p_1$ and $\displaystyle p_2$ in $\displaystyle P_2$ that span the kernel of $\displaystyle T$, and describe the range of $\displaystyle T$.
2. $\displaystyle kerT=\{p \in P_2:T(p)=0\}=\{p \in P_2(0)=0\}$
the range of T is the subspace of $\displaystyle R^2$ generated by (1,1).