Columns of a Change of Basis Matrix?

Question

I am currently learning transformation matrices with respect to different basis (A' = P^-1AP; probably described that wrong) and realized I needed to relearn change-of-basis matrices before I can even grasp this topic. First things first, I cannot figure out why a change-of-basis matrix (let's say P_A->B) has columns [a_i]_B and not the opposite ([b_i]_A). If that's a stupid question and I should just accept it, then okay... but it would really help me understand the topic if I could see, mathematically/logically, why the columns are set up that way. Thank you!

Answer 1

Remember that a change of basis matrix is used to multiply on the left side to a column vector.

So if you multiply 'P' with vector 'a' (which are coordinates of the basis A) you should get coordinates of the basis B.

P[a]$_{A}$ = [a]$_{B}$

What you do when you set up P with the columns [a$_{i}$]$_{B}$ is that you are asking "how can I express the vector 'a' according to the B basis?"

For example: A = {1, 1 + x, 1 + x + x$^2$} B = {1, x, x$^2$}

P$_A$$_B$ = ([a$_1$]$_B$[a$_2$]$_B$[a$_3$]$_B$)

[1,0,0]$_A$ = 1 = [1,0,0]$_B$

[0,1,0]$_A$ = 1 + x = [1,1,0]$_B$

[0,0,1]$_A$ = 1 + x + x$^2$ = [1,1,1]$_B$

P$_A$$_B$ = $$\begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{pmatrix} $$

Sorry english is not my first language and i'm also a student hope it helps!