Basis change matrix for polynomials.

Question

$\mathcal{B}$ 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} $

1. 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.

2. 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 $\mathcal{B}_1$.

3. 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 $\mathcal{B}_2$.

4. Applying the formula of basis change on a vector, find the results of question $1$.

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I have found:

1. $P = 10P_0 - 10P_1 - 7P_2 - 8P_3 $

The coefficients are: $10, -10, - 7$ and $- 8$.

2. $\mathcal{B_1}$ is a basis of $\mathbb{R}_3[X]$.

$P_{\mathcal{B}, \mathcal{B_1}} = \begin{pmatrix} 1 & 1 & 0 & 0 \\ 0 & -1 & 1 & 0 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & -1 \\ \end{pmatrix}$

3. $\mathcal{B_2}$ 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?

4. The basis change formula is: $B = P^{-1}.A.P$. I don't know how to find the results of question 1 using.

Answer 1

Please bear with me if you know the initial part of this answer.

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How do we write a matrix transformation? Given $(V,B_V)$ and $(W,B_W)$, where $V,W$ are vector spaces with bases $B_V,B_W$ respectively, and a linear transformation $T : V \to W$, how do we write the matrix for $T$, with respect to $B_V$ and $B_W$?

1. Arrange $B_V,B_W$ in some order.

2. Find where the elements of $B_V$ go under $T$.

3. Express the image vectors in terms of the basis $B_W$, and write the coefficients as columns one after the other to get a matrix.

For example, if $T : \mathbb R^2 \to \mathbb R^2$, both with standard bases, is given by $T(x) = -x$, then we find where $(1,0)$ goes : it goes to $(-1,0)$ in the standard basis, and then find where $(0,1)$ goes : it goes to $(0,-1)$ in the standard basis. Therefore, the matrix of $T$ will be : first column $(-1,0)$, second column $(0,-1)$. You can see that you will get minus the identity matrix if you put this together, for the matrix of $T$.

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Now, what is a change of basis matrix? Here is a nice interpretation :

The change of basis matrix from basis $A$ to basis $B$ (in a vector space $V$), is the matrix of the identity linear transformation from $(V,A)$ to $(V,B)$.

That is, find where all elements of $A$ go, and express them in basis $B$.

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What have you done? For the change of basis matrix from $B$ to $B_1$, this is the matrix of the identity map from $(V,B)$ to $(V,B_1)$. So, express the elements of $B$ in terms of the elements of $B_1$. You have done the opposite. You expressed $B_1$ in terms of $B$, when you wrote own your change of basis matrix.

The same with the third question.

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Consequently, we see that the change of basis matrices are the inverses of what you have written, because base conversion needs to go the other way. The right answer for b and c, therefore, is the inverse of the matrices you have written.

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Finally, for $d$, of course the given polynomial can be interpreted as an element of $(V,B)$. Now, to change the basis to $B_1$ we want to shift to $(V,B_1)$, so simply multiply the change of basis matrix you have with you, and the vector expressed in $B_1$.

That is, the inverse of the $P_{B,B_1}$ which you got, when multiplied with $[0,3,-1,8]^T$, gives you $[10,-10,-7,-8]$. You can check this is right.

Answer 2

The base change matrices are correct.
To prove the given polynomials form a basis, just note that the determinants of these matrices are nonzero.

For question 4, the formula $P^{-1}AP$ applies only to a matrix $A$.
Here we only have a vector $b$ with $[b]_\mathcal B =(0,3,-1,8)^T$.
To express $[b]_{\mathcal B_1}$, calculate $P^{-1}$ (e.g. by manually expressing the standard basis elements by $P_i$'s) and then apply/verify $$[b]_{\mathcal B_1}=P^{-1}[b]_\mathcal B$$ Note that, for a base change matrix $P:\mathcal B\leadsto\mathcal B_1$ and a square matrix $A$, the columns of $AP$ are just $A$ applied to the elements of $\mathcal B_1$, coordinated in $\mathcal B$, so $P^{-1}$ of this will contain the $\mathcal B_1$-coordinates of the same vectors.