My linear algebra textbook presents a technique for computing the transition matrix from an old basis B' to a new basis B. Apparently if you set up an augmented matrix with B on the left and B' on the right, then put B into RREF, the resulting matrix on the right after those row operations will be the transition matrix from B' to B. The book doesn't explain why this works and I can't figure out why myself. Can anyone help me out? Thanks.