If I have a linear transformation $T$ and the basis $b$ and $a$, when I have $[T]_{b}^{a}$ It means that I'm doing the transformation starting on the basis $a$ and the result is given with respect of basis $b$?!
If so, it is correct to apply first the transformation with respect to basis $a$ : $[T]_{a}^{a}$ and then apply another transformation, $[Q]_{b}^{a}$ for example, that change the basis from $a$ to $b$?
What I am trying to get is that the relation below is true or not: $$ [T]_{b}^{a} = [Q]_{b}^{a}\cdot [T]_{a}^{a} $$
Thanks.
Note that if you want $[Q]^a_b$ to take a coordinate-vector with respect to $a$ and produce a coordinate-vector with respect to $b$, then the transformation $Q$ itself "shouldn't do anything". That is, for any vector $x$, we should have $Q(x) = x$. That is, $Q$ should be the identity operator. Typically, $I$ is used to denote this identity operator.
With that being said, it is indeed true that $$ [T]^a_b = [I]^a_b[T]^a_a $$