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.

  • $\begingroup$ okay, but i have an dude because i don't know how is the 4x4 matrix, because scalar confuse me. @Lem.ma $\endgroup$ – Bvss12 Apr 24 '16 at 3:44
  • $\begingroup$ 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} $ $\endgroup$ – Bvss12 Apr 24 '16 at 3:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.