Linear maps that have the same matrix regardless of the bases chosen for domain and codomain Question: Other than the zero map, what linear map has the same matrix $A_{E,F}$ with respect to all $E$ and $F$?
For linear map $T:\mathbb{R}^n \rightarrow  \mathbb{R}^n$, given a basis $E$ for domain and basis $F$ for codomain, I can find a unique corresponding matrix $A_{E,F}$, where $A_{E,F}$ generally depends on $E$ and $F$.
Note that for any $E$ and $F$, the matrix corresponding to the zero map is always the zero matrix, since the map sends all vectors in $E$ to $\mathbf{0}$, and $\mathbf{0}$ can only be represented by $(0,...,0)$ with respect to any basis $F$.

Let $A$ be matrix of $T$ with basis on both the range and domain as the columns of the identity matrix $I$.
Then $A/2 $ the matrix with the same basis on the domain and $2I$ as the basis on the range.
So, $A/2 = A$, or $A=0$.
