For a linear transformation $T:R^n \to R^m$ defined by $T(x)=Ax$ where $A$ is some $m \times n$ matrix, to my understanding, if some arbitrary basis is used for both $R^n$ ($\{v_1 ... v_n \}$) and $R^m$ ($\{u_1 ... u_m \}$), the ith column of $A$ should be the vector $T(v_i) = \begin{bmatrix} c_{1i} \\ c_{2i} \\ \vdots \\ c_{mi}\end{bmatrix} = \sum_{j=1}^mc_{ji}u_j$ where the c's are the coordinates defined in terms of the basis $\{u_1 ... u_m \}$ for $R^m$.
For the subspaces $Col(A), Row(A), Null(A), Null(A^T)$, what basis is used to define the coordinates of the vectors in these spaces? I'm particularly stumped on $Row(A)$ and $Null(A^T)$ since I can't see a relation between it and the domain / range of the transformation for which we defined a basis.