When I have a linear operator $T:V\rightarrow W$ and images for some ordered basis $B=(b_1,\dots,b_n)$ of $V$, what do I have exactly if I put those images in a matrix? Is it by default:
$[T]^B_E = \begin{bmatrix} T(b_1) & \dots & T(b_n) \end{bmatrix}^B_E$
where the output is according to the standard basis of $W$? What are those images if I don't take their coordinate vectors?