Relative Matrix of Linear Transformation/Examples/Differentiation

Example of Relative Matrix of Linear Transformation
Let $\map {P_3} \R$ and $\map {P_4} \R$ denote the sets of real polynomials of degree $3$ and $4$ respectively.

Let $\sequence {I^J}_{0 \mathop \le j < n}$ denote the ordered bases of $\map {P_3} \R$ and $\map {P_4} \R$ where $n = 3$ and $n = 4$ respectively.

Let $D: p \to D p$ denote the operation of differentiation on a real polynomial $p$.

Let $\mathbf D := \sqbrk {D; \sequence {I^J}_{0 \mathop \le j < 3}, \sequence {I^J}_{0 \mathop \le j < 4} }$ denote the relative matrix of the linear transformation that is $D$.

Then:
 * $\mathbf D = \begin {bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 \end {bmatrix}$