# Transition matrix of polynomial.

Good night, i need help with this.

Find the transition matrix that goes from the basis W to the basis $\left\{ 2,1-2x,x^{2}-1,x^{3}-x^{2}+x\right\}$

I found a basis for W, $\left\{ 2,x,x^{2}+1,x^{3}\right\}$ and i work in a linear combination of basis W to V and i make a system, but then I had problems with the system

$\begin{cases} 2=\alpha_{1}\left(2\right)+\alpha_{2}\left(1-2x\right)+\alpha_{3}\left(x^{2}-1\right)+\alpha_{4}\left(x^{3}-x^{2}+x\right)\\ x=\alpha_{1}\left(2\right)+\alpha_{2}\left(1-2x\right)+\alpha_{3}\left(x^{2}-1\right)+\alpha_{4}\left(x^{3}-x^{2}+x\right)\\ x^{2}+1=\alpha_{1}\left(2\right)+\alpha_{2}\left(1-2x\right)+\alpha_{3}\left(x^{2}-1\right)+\alpha_{4}\left(x^{3}-x^{2}+x\right)\\ x^{3}=\alpha_{1}\left(2\right)+\alpha_{2}\left(1-2x\right)+\alpha_{3}\left(x^{2}-1\right)+\alpha_{4}\left(x^{3}-x^{2}+x\right) \end{cases}$

All you need to do now is to organize the four sets of the four $\alpha$'s that you will find from your four systems of equations (do you see why each identity you wrote is a system of equations?) into a $4 \times 4$ matrix $A$.
Finally, whether it's $A$, $A^T$, $A^{-1}$, or $(A^T)^{-1}$ that is the answer you're looking for depends on your definition of transition matrix.
• i work in a matrix, $\begin{bmatrix}2 & 1-2x & x^{2}-1 & x^{3}-x^{2}+x & =2\\ 2 & 1-2x & x^{2}-1 & x^{3}-x^{2}+x & =x\\ 2 & 1-2x & x^{2}-1 & x^{3}-x^{2}+x & =x^{2}\\ 2 & 1-2x & x^{2}-1 & x^{3}-x^{2}+x & =x^{3} \end{bmatrix}$ – Bvss12 Apr 24 '16 at 3:48