The algebraic topology room has so many folks. Maybe we can have some too?

Or maybe I should make one called Group, Ring, Field and Galois theory

Say I have two sets of basis vectors $E = {e_1, e_2, e_3}$ and $E' = {e_1', e_2', e_3'}$ for a vector space $V$. If I express the $e_i'$ as linear combinations of the $e_i$ and write the coefficients as columns of a matrix, does that matrix represent the change of basis matrix for the linear map $e_i \mapsto e_i'$ and if we multiply it by a vector's (say $v = a_1 e_1 + a_2 e_2 + a_3 e_3$) coordinate $(a_1,a_2,a_3)^{T}$, do I then get the co-ordinate of $v$ in the basis $E'$?

Why don't you just test it?

I've tried it out but it seems that it's not the case.

I found the matrix of $e_i \mapsto e_i'$ and multiplied it by the coordinate of $v$ but it seems that the inverse of that matrix multiplied by the coordinate gives $v$ as a linear combination of the $e_i'$, @Bye_World.

If you construct your matrix with columns of $e_i'$ in terms of $e_i$ then you'll get the change of basis matrix from E' to E, not the other way around.

Would that then be the matrix corresponding to the linear map $e_i \mapsto e_i'$?

@Khallil

Maybe this answer I gave the other day will help you: math.stackexchange.com/questions/1460141/…