Find the transition matrix from $(1+x^2,2x,1)$ to $(1,x,x^2)$

Question

This is a question with its solution i was wondering if it is correct and if it is or it is not can someone explain what is going on and why do we need $B=P^{-1} A P$

$B$ and $A$ are given along with beta and beta'

Answer 1

The transition matrix can't be singular since, as is fairly easy to see, you have two bases.

Looks like the problem occurs in the last column, where you should have $(1,0,0)^t$.

It turns out that that the way the transition matrix $P$ works is that $B=P^{-1}AP$ will be the matrix $A$ expressed in the new basis. That's precisely because $P$ takes vectors written in the basis $\beta'$ and gives them back written in terms of $\beta$. ($P^{-1}$ of course does the reverse.)

Answer 2

Everything was going smoothly until the third equation. Why did you write $I(1)=0? Check that equation.

$$ 1= 1\cdot 1 + 0 \cdot x + 0 \cdot x^2 $$

So your last vector would be $(1,0,0)^t$ making the matrix ($P$) non-singular. It is relevant that $P$ is non-singular as you need to invert it (get $P^{-1}$).