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**Runty** This is pretty difficult stuff, so be sure to read all of it.

Suppose that $\displaystyle V$ and $\displaystyle W$ are vector spaces over a field $\displaystyle F$.

$\displaystyle F[x]$ is the vector space (over $\displaystyle F$) of polynomials with coefficients in $\displaystyle x$, i.e.

$\displaystyle F[x]=\{a_0+a_1x+...+a_nx^n:a_0,...,a_n\in F, n\geq 0\}$.

Fix a positive integer $\displaystyle m$. For this, you may use without proof that the subset $\displaystyle W\subset F[x]$ of all polynomials of degree $\displaystyle m$ or less is a subspace of $\displaystyle F[x]$ and that this subspace has two bases

$\displaystyle B=\{1,x,x^2,...,x^m\}$ and $\displaystyle B'=\{1,x,x^2,...,x^{m-1},x^{m-1}+x^m\}$.

**(a)** Determine the basechange matrix $\displaystyle P\in F^{m+1\times m+1}$ such that $\displaystyle B'=BP$ and compute the determinant of $\displaystyle P$.

A hint was provided as follows: Determine $\displaystyle P$ by specifying its matrix entries $\displaystyle p_{ij}\in F$.