In doing some analysis in finite dimensional space, the following question arose about the link between columns of a matrix under different basis.
More precisely, consider $(e_i)$ the canonical basis in $\mathbb{R}^{d}$ and $(\tilde{e}_i))$ another basis. We denote by $H$ the matrice from $(e_i)$ to $(\tilde{e}_i)$. As a consequence we have the following relation : $\tilde{e}_i = \sum_{k=1}^{d} h_{k,i}e_k$ for all $i$.
Now if I consider a linear map $f :\mathbb{R}^{d}\to\mathbb{R}^{d}$, $A$ his matrix representation in the canonical basis and $\tilde{A}$ his matrix representation in the basis $(\tilde{e}_i)$ do we have that
$$ A\tilde{e}_i = \tilde{A}e_i $$
?
I have try to prove it using my knowledge on this question but I did not managed to do it. I am stuck at this step
$$ A\tilde{e}_i = H\tilde{A}H^{-1}\tilde{e}_i =H\tilde{A}e_i $$
Thus I am wondering if this result is true, and if not, under what assumption could it be.
Thank you a lot !