I was reading "Elementary Linear Algebra Applications" by Howard Anton, and in section 4.6, Change of Basis, it talks about finding the transition matrix, $P$, from an old basis $B$ to to another new basis $B'$. I understand that if the transition matrix satisfies $P_{B \to B'} B = B'$, then the the transition matrix can be found with $P_{B \to B'} = \{ [u_1']_B | [u_2']_B |\cdots | [u_n']_B \}$, where $u_i$ is the $i$th basis/column in $B'$ and $[v]_B$ is the vector's coordinates with respect to basis $B$.
The textbook then says this:
I don't understand how they got step 3, and i couldn't find a proof in the textbook... i don't see how simultaneously solving n systems is connected to this... what is the "same coefficient matrix" in the n systems? An explanation + proof would be greatly appreciated!
Row operations correspond to left multiplication by a matrix. Thus, performing row operations that turn $B'$ into $I$ is the same as left multiplying by $B'^{-1}$. If we do the same row operations to $B$, the result is $B'^{-1}B$.
Let $e_i$ be the vector $\left(\begin{matrix}0 \\ ... \\ 1 \\ ... \\ 0 \end{matrix} \right)$ with the $1$ at the $i$th position. Then $B e_i$ is the $i$th column of $B$, so $B$ changes the $B$ basis to the standard basis. Then, left multiplying by $B'^{-1}$ changes the standard basis to the $B'$ basis, so $B'^{-1}B$ changes from the $B$ basis to the $B'$ basis.