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 =_=