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# Find a COB Matrix for Cubic Polynomials

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Let $V$ be the vector space of polynomials of degree at most $3$.

The lists:

$$\mathcal{B}_1=\{1,x,x^2,x^3\}$$

...and:

$$\mathcal{B}_2=\{1+x,1-x,x^2+x^3,2x^3\}$$

...are two bases of $V$.

Which of the following is the matrix $M$ such that:

$$M[f]_{\mathcal{B}_1}=[f]_{\mathcal{B}_2}$$

...for all $f\in V$?

A

$\begin{pmatrix}\cfrac{1}{2}&\cfrac{1}{2}&0&0\\\ \cfrac{1}{2}&-\cfrac{1}{2}&0&0\\\ 0&0&1&-\cfrac{1}{2}\\\ 0&0&0&\cfrac{1}{2}\end{pmatrix}$

B

$\begin{pmatrix}\cfrac{1}{2}&\cfrac{1}{2}&0&0\\\ \cfrac{1}{2}&-\cfrac{1}{2}&0&0\\\ 0&0&1&0\\\ 0&0&-\cfrac{1}{2}&\cfrac{1}{2}\end{pmatrix}$

C

$\begin{pmatrix}1&1&0&0\\\ 1&-1&0&0\\\ 0&0&1&0\\\ 0&0&1&2\end{pmatrix}$

D

$\begin{pmatrix}1&1&0&0\\\ 1&-1&0&0\\\ 0&0&1&1\\\ 0&0&0&2\end{pmatrix}$