when we say that t A is an nxn matrix whose columns are given by the vectors in S where S is a basis for R^n,
does it mean that A is a transition matrix from S to S'' where S'' is the standard basis in R^n?
thanks!
You are quite right.
To elaborate, if $\displaystyle S=\{v_1,v_2,\ldots,v_n\}$ is the given basis for $\displaystyle \mathbb{R}^n$, and if $\displaystyle A$ is the matrix, whose $\displaystyle i$th column is the basis vector $\displaystyle v_i$, then if $\displaystyle [a]_S$ is the coordinates of the vector $\displaystyle a\in\mathbb{R}^n$ with respect to the (ordered) basis $\displaystyle S$, then
$\displaystyle A[a]_S = a,$
and the vector $\displaystyle a$ can be considered as itself written in the standard basis.