Question

Find the change of coordinates matrix from B = [3,2,8;-1,0.-2;4,-5,7] to the standard basis in R^3

Answer 1

If you have two bases, B and B' in R^n, then the transition matrix, P^(-1) from B to B' can be found by using Gauss-Jordan elimination on the n x 2n matrix, \[[B':B] \rightarrow [I_n:P^{-1}]\]Since you're moving from B to I_3, you're starting with\[[I_3:B]\]which is in the form\[[I_3:P^{-1}]\]already. So the change of coordinates matrix would be B.