Transformation matrix relative to the standard basis

Question

Consider the vector space P3 with its standard basis E = {1,t,t^2,t^3} Consider the linear transformation T(p(t) = p''(t)

Find the matrix [T]E

Answer 1

Reminder. Let $E$ be a finite-dimensional vector-space, $\underline{e}:=(e_1,\cdots,e_n)$ a basis of $E$ and $T\colon E\rightarrow E$ a linear transformation. For all $i\in\{1,\cdots,n\}$, there exists $(\lambda_{i,j})_{j\in\{1,\cdots,n\}}$ such that: $$T(e_i)=\sum_{j=1}^n\lambda_{i,j}e_j.$$ With these notations, one has: $$\textrm{mat}(T,\underline{e})=(\lambda_{i,j})_{1\leqslant i,j\leqslant n}.$$

Hint. Compute $T(1),T(t),T(t^2),T(t^3)$ and decompose these elements in the basis $\{1,t,t^2,t^3\}$.