$[T ]^{b}_{b'}$
$[T ]_{b'}$
[ ]$_{b}$
Lets suppose $T$ is s a linear map from $V$ to $W$, and $b'$ and $b$ are bases in $V$ and $W$ respectively. I would like to know what this notation means in each case and how it is read (like "$T(v)$" is read aloud as: "T of v"). I have a shakey idea of what it means but I'd like a definitive explanation so that I can have a solid understanding.
I'd also be very grateful if someone can also point me to some resource that covers matrix representation of linear maps that uses this notation; as I couldn't find any --again, I don't know what this style of notation is called.
$[T]$ is the matrix representative of the linear map $T$ in whatever basis is obvious, usually the standard basis. $[T]_{b}^{b'}$ is the matrix representative of $T$ which takes coordinates in the basis $b$ and gives coordinates in the basis $b'$.