What does $[v]_e$ mean in linear transformations?

Question

$[v]_e$ keeps on popping up in my linear transformations chapter at university and I have no clue what it means, in my homework it asks find the coordinate vector of $[v]_e$ when v=(1,0,2) for the linear transformation T(x,y,z)=(x+2y−z,x+z,4x−4y+5z), I have already found it's matrix trans wrt the standard basis however how do I go from here to answer this question?

Answer 1

I expect that this notation is explained somewhere earlier in the textbook that you’re using, if not in that very chapter. Be that as it may, $[v]_{\mathcal B}$ stands for the coordinates of the vector $v$ relative to the ordered basis $\mathcal B$, not to be confused with the vector $v$ itself. This notation distinguishes between vectors and their coordinate tuples relative to some basis, even if the vectors themselves are tuples of scalars. The subscript $e$ could indicate the standard basis or some other basis that’s given in context.