01 - Isomorphism between Linear Maps and Matrices

Isomorphism between Linear Maps and Matrices

Map $T\mapsto {}_{{\mathcal{B}}^{\prime }}[T{]}_{\mathcal{B}}$ is an isomorphism from $\mathrm{Hom}(V,W)$ to ${\mathbf{M}}_{m\times n}(\mathbf{F})$

That is it takes composition of maps to multiplication of matrices

Special Case

If $T:V\to V$ and $\mathcal{B}$, ${\mathcal{B}}^{\prime }$ are two different bases with ${}_{{\mathcal{B}}^{\prime }}[\mathrm{Id}{]}_{\mathcal{B}}$ the change of basis matrix then

$${}_{{\mathcal{B}}^{\prime }}[T{]}_{{\mathcal{B}}^{\prime }}={}_{{\mathcal{B}}^{\prime }}[\mathrm{Id}{]}_{\mathcal{B}}{}_{{\mathcal{B}}^{\prime }}[T{]}_{\mathcal{B}}{}_{\mathcal{B}}[\mathrm{Id}{]}_{{\mathcal{B}}^{\prime }}$$

with

$${}_{\mathcal{B}}[\mathrm{Id}{]}_{{\mathcal{B}}^{\prime }}{}_{{\mathcal{B}}^{\prime }}[\mathrm{Id}{]}_{\mathcal{B}}={}_{\mathcal{B}}[\mathrm{Id}{]}_{\mathcal{B}}=I$$