$T : \Bbb P_n \to \Bbb P_n$ given by $T(f(t)) = \dfrac{df(t)}{dt}$, where $\Bbb P_n$ is space of polynomials with degree $n$.
The matrix representation of this linear transformation (basis as $\{1, .., t^n\}$) would be $$T(\mathbf v) = \begin{bmatrix} 0&1&0&\cdots&0\\ 0&0&2&\cdots&0\\ 0&0&0&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&n\\ 0&0&0&\cdots&0 \end{bmatrix} \mathbf v$$
where $\bf v$ is $\Bbb R^n$ column vector representation of $f(t)$, like if $f(t) = \sum_{k = 0}^n \alpha_k t^k$ then $v = (\alpha_0, ... , \alpha_n)^T$.
If I am not wrong then the $T$ in matrix representation is not same as $T$ of the original question. The $T$ of the original question is $T : \Bbb P_n \to \Bbb P_n$ whereas the $T$ in the matrix representation is $T : \Bbb R^n \to \Bbb R^n$.
So why we call the $T$ in matrix representation same as the $T$ in original question ?
Is matrix represention of a linear transformation $R : V \to W$ just a way to write the same information as some other linear tranformation $S : \Bbb F^n \to \Bbb F^n$ ?