Suppose I have a linear transformation $T: V\longrightarrow V$, where $V$ is a finite dimensional vector space. Let $A$ be the matrix representation of $T$ under the basis $\beta$ of $V$.
Are there linear transformations $T_1, T_2, T_3, T_4, \dots$ such that $T= \sum_i T_i$? Or is there is a way to approach that goal?
Also, will it be true that $A$ is the sum of the matrix representation of each of $T_1, T_2, T_3, T_4, \dots$ under basis $\beta$?
Thanks in advance.