Problem: Let $W$ be the vector space of all real polynomials of degree at most $3$. Define \[ T:W\to W,~ T(p(x)) = p^\prime (x). \] Then find out the matrix of $T$ with respect to the basis $\left\{ 1,x,x^2 , x^3 \right\} $.
Solution: It is easy to check that the given map is a linear transformation. Observe that \begin{align*} T(1) & = 0 \implies 0\cdot 1+ 0\cdot x + 0\cdot x^2 + 0\cdot x^3 \\ T(x) & = 1 \implies 1\cdot 1+ 0\cdot x + 0\cdot x^2 + 0\cdot x^3 \\ T\left( x^2 \right) & = 2x \implies 0\cdot 1+ 2\cdot x + 0\cdot x^2 + 0\cdot x^3 \\ T\left( x^3 \right) & = 3x^2 \implies 0\cdot 1+ 0\cdot x + 3\cdot x^2 + 0\cdot x^3. \end{align*} Therefore, the matrix of the given linear transformation will be \[ [T] = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 \\ \end{pmatrix}. \]