Question

Find the transition matrix representing the change
of coordinates on P3 from the ordered basis
[1, x, x2] to the ordered basis
[1, 1 + x, 1 + x + x2]

WHY WE CANNOT FIND THE TRANSITION MATRIX FROM [1, x, x2] to the ordered basis
[1, 1 + x, 1 + x + x2] BECAUSE THE SOLUTION IS USING THE REVERSE AND TAKE THE INVERSE

Step 1 of 3 The objective is to find the transition matrix represent the change of coordinates on P from the ordered basis 1,x.rto 1,1+x.1+x+x Comment Step 2 of 3A We want to change from the ordered basis |1,x.x*to 1,1+x,1+x+x Since, 1,x,xis the ordered basis for P , easy to write transition matrix fronm 1,1+x,1+x+x to 1,x.x That is, 1=1-1,+ 0 . x + 0.x2 1+x=1-1+1.x+0.x The transition matrix is The inverse of S will be given transition matrix from |1.x,rto1,1+x,1+x+x

Answer 1

There are two ways of finding the transition matrix from $[1,x,x^2]$ to   $[1,1+x,1+x+x^2]$

First is to find the inverse of , and

second one is repeating the same steps as done to find .

So the matrix is

It is easy to verify that $R=S^{-1}$