$\mathcal{B}$ is the canonical basis of $\mathbb{R}_3 [X]$.
We have the polynomials:
$\begin{cases} P_0 = 1, \\ P_1 = 1 - X, \\ P_2 = X - X^2, \\ P_3 = X^2 - X^3. \\ \end{cases} $ and $ \begin{cases} Q_0 = 1, \\ Q_1 = 1 + X, \\ Q_2 = 1 + X + X^2, \\ Q_3 = 1 + X + X^2 + X^3. \end{cases} $
Prove that $P = 3X - X^2 + 8X^3$ can be expressed as linear combination of the polynomials $P_0,P_1,P_2,P_3$ and determine the coefficients.
Prove that the $\mathcal{B_1} = (P_0, P_1, P_2, P_3)$ is a basis of $\mathbb{R}_3 [X]$ and determine the matrix $P_1$ of change of basis from $\mathcal{B}$ to $\mathcal{B}_1$.
Prove that $\mathcal{B}_2 = (Q_0, Q_1, Q_2, Q_3)$ is a basis of $\mathbb{R}_3[X]$ and determine the matrix $P_2$ of change of basis from $\mathcal{B}$ to $\mathcal{B}_2$.
Applying the formula of basis change on a vector, find the results of question $1$.
I have found:
- $P = 10P_0 - 10P_1 - 7P_2 - 8P_3 $
The coefficients are: $10, -10, - 7$ and $- 8$.
- $\mathcal{B_1}$ is a basis of $\mathbb{R}_3[X]$.
$P_{\mathcal{B}, \mathcal{B_1}} = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 0 & -1 & 1 & 0 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & -1 \\ \end{pmatrix}$
- $\mathcal{B_2}$ is a basis of $\mathbb{R}_3[X]$.
$P_{\mathcal{B}, \mathcal{B_2}} = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}$
Are basis change matrices correct?
- The basis change formula is: $B = P^{-1}.A.P$. I don't know how to find the results of question 1 using.