1. ## standard basis

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!

2. You are quite right.

To elaborate, if $S=\{v_1,v_2,\ldots,v_n\}$ is the given basis for $\mathbb{R}^n$, and if $A$ is the matrix, whose $i$th column is the basis vector $v_i$, then if $[a]_S$ is the coordinates of the vector $a\in\mathbb{R}^n$ with respect to the (ordered) basis $S$, then

$A[a]_S = a,$

and the vector $a$ can be considered as itself written in the standard basis.