Q1: Let $V, W$ be two finite-dimensional $\mathbb{F}$-vector spaces and $\ell: V \rightarrow W$ be a linear map. Is it possible that two different matrices represent $\ell$?

A1: Yes, but for some linear maps there is only one matrix representation.

A2: Yes, one can always find two different maps when one chooses different bases on the vector spaces.

A3: No, it’s never possible.

A4: No, this only works if $V=W$.

Q2: Let $V, W$ be two finite-dimensional $\mathbb{F}$-vector spaces and $\ell: V \rightarrow W$ be a linear map. We denote the change-of-basis matrices by $T_{\mathcal{B} \leftarrow \widetilde{\mathcal{B}}}$ and $T_{\widetilde{\mathcal{C}} \leftarrow \mathcal{C}}$ for bases $\mathcal{B}, \widetilde{\mathcal{B}}, \widetilde{\mathcal{C}}, \mathcal{C}$. What is correct?

A1: $\ell_{\widetilde{\mathcal{B}} \leftarrow \widetilde{\mathcal{C}}} = T_{\widetilde{\mathcal{B}} \leftarrow \mathcal{B}} \ell_{{\mathcal{B}} \leftarrow {\mathcal{C}}} T_{\mathcal{C} \leftarrow \widetilde{\mathcal{C}}} $

A2: $\ell_{\widetilde{\mathcal{B}} \leftarrow \widetilde{\mathcal{C}}} = T_{\widetilde{\mathcal{C}} \leftarrow \mathcal{B}} \ell_{{\mathcal{B}} \leftarrow {\mathcal{C}}} T_{\mathcal{B} \leftarrow \widetilde{\mathcal{C}}} $

A3: $\ell_{\widetilde{\mathcal{B}} \leftarrow \widetilde{\mathcal{C}}} = T_{\widetilde{\mathcal{C}} \leftarrow \mathcal{B}} \ell_{{\mathcal{B}} \leftarrow {\mathcal{C}}} T_{\mathcal{B} \leftarrow \mathcal{C}} $

A4: $\ell_{\widetilde{\mathcal{B}} \leftarrow \widetilde{\mathcal{C}}} = T_{\mathcal{C} \leftarrow \mathcal{B}} \ell_{{\mathcal{B}} \leftarrow {\mathcal{C}}} T_{\mathcal{B} \leftarrow \mathcal{C}} $