09 - Change of Basis Theorem

Change of Basis Theorem

Let $V$ be a finite-dimensional vector space over $\mathbf{F}$ with ordered $\mathcal{V},{\mathcal{V}}^{\prime }$
Let $W$ be a finite-dimensional vector space over $\mathbf{F}$ with ordered bases $\mathcal{W},{\mathcal{W}}^{\prime }$

Let $T:V\to W$ be a linear map
Then

$${}_{{\mathcal{W}}^{\prime }}{T}_{{\mathcal{V}}^{\prime }}=({}_{{\mathcal{W}}^{\prime }}{I}_{\mathcal{W}})({}_{\mathcal{W}}{T}_{\mathcal{V}})({}_{\mathcal{V}}{I}_{{\mathcal{V}}^{\prime }})$$

Follows from 08 - Matrix Representation of a Composition of Linear Maps =_=

Change of Basis Theorem V2

Let $V$ be a finite-dimensional vector space over $\mathbf{F}$ with ordered basis $\mathcal{V},{\mathcal{V}}^{\prime }$
Let $T:V\to V$ be a linear map
Then

$${}_{{\mathcal{V}}^{\prime }}{T}_{{\mathcal{V}}^{\prime }}={(}_{{\mathcal{V}}^{\prime }}{I}_{\mathcal{V}}){(}_{\mathcal{V}}{T}_{\mathcal{V}}){(}_{\mathcal{V}}{I}_{{\mathcal{V}}^{\prime }})$$

If we set $A={}_{{\mathcal{V}}^{\prime }}{T}_{{\mathcal{V}}^{\prime }},B={}_{\mathcal{V}}{T}_{\mathcal{V}},P={}_{\mathcal{V}}{I}_{{\mathcal{V}}^{\prime }}$
Then

$$A=P^{-1}BP$$

Special case from the proof above :)