Write formula for $[T]$ using change of basis formulas

I'm having a hard time with this questions here. So I need to give an expression for $[T]^δ_γ$ in terms of other given matrices found. So the matrices I have are $[I]_γ^a$, $[I]^β_δ$, $[T]^β_α$, $[T]^δ_γ$. I've trying to manipulate change of basis formulas in order to get an expression for $[T]^δ_γ$, but I can't seem to figure it out. Please provide me with some direction.

Thanks!!


The matrix you seek is $\Big([I]_{\delta}^{\beta}\Big)^{-1}[T]_{\alpha}^{\beta}[I]_{\gamma}^{\alpha}$. This is because $$\begin{eqnarray*}\Big([I]_{\delta}^{\beta}\Big)^{-1}[T]_{\alpha}^{\beta}[I]_{\gamma}^{\alpha}[x]_{\gamma} &=& \Big([I]_{\delta}^{\beta}\Big)^{-1}[T]_{\alpha}^{\beta}[x]_{\alpha} \\ &=& \Big([I]_{\delta}^{\beta}\Big)^{-1}[Tx]_{\beta} \\ &=& [I]_{\beta}^{\delta}[Tx]_{\beta} \\ &=&[Tx]_{\delta}\end{eqnarray*}$$ The above calculation demonstrates how $\Big([I]_{\delta}^{\beta}\Big)^{-1}[T]_{\alpha}^{\beta}[I]_{\gamma}^{\alpha}$ maps the coordinate vector $[x]_{\gamma}$ to the coordinate vector $[Tx]_{\delta}$, so it must be the case that $$\Big([I]_{\delta}^{\beta}\Big)^{-1}[T]_{\alpha}^{\beta}[I]_{\gamma}^{\alpha}=[T]_{\gamma}^{\delta}$$