Changing Coordinates of Endomorphic Maps

Theorem

• With $T:V\to V$
• $[T{]}_{\alpha }^{{\alpha }^{\prime }}=[I{]}_{\alpha }^{{\alpha }^{\prime }}[T{]}_{\alpha }^{\alpha }[I{]}_{{\alpha }^{\prime }}^{\alpha }=([I{]}_{{\alpha }^{\prime }}^{\alpha }{)}^{-1}[T{]}_{\alpha }^{\alpha }[I{]}_{{\alpha }^{\prime }}^{\alpha }$

Proof

1. Taking $V=W,\beta =\alpha ,{\beta }^{\prime }={\alpha }^{\prime }$
2. We get,