basic matrix question about change of basis matrices.

Question

we have that $A=[T]_{a}^{a}$ and i believe that $ [T]_{a}^{a}=[I]^{a}[T][I]_{a}$

i am wondering what the justification is it kind of feels like taking something in the standard basis converting it into the a basis then acting on it with the T matrix. Then i ask for the solutions to that in the a basis?

Answer 1

We have that

$$w=T(v)\iff w=[T]v$$

and

$$w=[I]_aw_a \quad v=[I]_av_a $$

thus

$$w=[T]v \iff [I]_aw_a=[T][I]_av_a \iff w_a=[I]^a[T][I]_av_a $$

therefore

$$[T]_{a}^{a}=[I]^{a}[T][I]_{a}$$