Transpose of the matrix of a linear map

Question

Given a linear map $\phi: V \rightarrow W,$ suppose $A$ is the matrix of the linear map with respect to the bases $v_1, \dots, v_n$ and $w_1, \dots, w_m.$ Is there any easy way we can modify the bases such that the matrix of the linear map becomes $A^{\top}$ instead?

Answer 1

In general no, since $A$ and $A^\top$ have different shapes (unless $m=n$).

Answer 2

No, not with the way that matrix multiplication is defined, the change of basis matrix is unique. Some thoughts:

• If $m=n$, then you can find another set of basis vectors whose change of basis is $A^T$, but it's not obvious how they're related to the $v$'s and $w$'s.

• If you dualize everything, and treat your vectors as row vectors (or dual vectors), then your change of basis matrix would be $A^T$, but you would also be multiplying on the left, so that might be cheating.