# Change of basis computation gone wrong...

## Homework Statement

Consider the real-vector space of polynomials (i.e. real coefficients) $f(x)$ of at most degree $3$, let's call that space $X$. And consider the real-vector space of polynomials (i.e. real coefficients) of at most degree $2$, call that $Y$. And consider the linear transformation $A$ from $X$ to $Y$ defined by the following:$$A(f) = 2f' - (x+1)f''.$$Let $$[A]_{(1,x,x^2,x^3)}^{(1,x,x^2)}$$be the matrix of $A$ with respect to the bases $(1, x, x^2, x^3)$ of $X$ and $(1, x, x^2)$ of $Y$, let$$[A]_{(1,x+1,(x+1)^2,(x+1)^3)}^{(1,x+1,(x+1)^2)}$$be the matrix of $A$ with respect to the bases $(1,x+1,(x+1)^2,(x+1)^3)$ of $X$ and $(1,x+1,(x+1)^2)$ of $Y$, let $B$ be the change of basis matrix from $(1,x,x^2,x^3)$ to $(1,x+1,(x+1)^2,(x+1)^3)$, and let $C$ be the change of basis matrix from $(1, x, x^2)$ to $(1,x+1,(x+1)^2)$.

Problem. Show that$$[A]_{(1,x+1,(x+1)^2,(x+1)^3)}^{(1,x+1,(x+1)^2)} = C^{-1}[A]_{(1,x,x^2,x^3)}^{(1,x,x^2)}B.$$

## Homework Equations

This category is a bit silly.

## The Attempt at a Solution

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Let's first calculate$$[A]_{(1,x,x^2,x^3)}^{(1,x,x^2)}.$$Assume $f(x)=a+bx+cx^2+dx^3$, then$$A(f)=2f' - (x+1)f''=2(b+2cx+3dx^2)-(x+1)(2c+6dx)=$$$$=2b+4cx+6dx^2-2cx-6dx^2-2c-6dx=(2b-2c)+(2c-6d)x,$$
therefore
$$[A]_{(1,x,x^2,x^3)}^{(1,x,x^2)}.=\begin{pmatrix}0&2&-2&0\\0&0&2&-6\\0&0&0&0\end{pmatrix}.$$Okay, now let's do$$[A]_{(1,x+1,(x+1)^2,(x+1)^3)}^{(1,x+1,(x+1)^2)},$$which is similar. So we begin with $f(x) = a + b(x+1) + c(x+1)^2 + d(x+1)^3$, then$$A(f) = 2f' - (x+1)f'' = 2(b + 2c(x+1) + 3d(x+1)^2) - (x+1)(2c + 6d(x+1)) = 2b + 2c(x+1).$$So$$[A]_{(1,x+1,(x+1)^2,(x+1)^3)}^{(1,x+1,(x+1)^2)}.=\begin{pmatrix}0&2&0&0\\0&0&2&0\\0&0&0&0\end{pmatrix}.$$Okay, onto $B$, $C$, $C^{-1}$. I calculated that$$B = \begin{pmatrix} 1 & -1 & 1 & -1 \\ 0 & 1 & -2 & 3 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 1\end{pmatrix},$$
$$C = \begin{pmatrix} 1 & -1 & 1 \\ 0 & 1 & -2 \\ 0 & 0 & 1\end{pmatrix},$$
$$C^{-1} = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1\end{pmatrix}.$$However, when I plug everything into my original equation which I want to verify, i.e.$$[A]_{(1,x+1,(x+1)^2,(x+1)^3)}^{(1,x+1,(x+1)^2)} = C^{-1}[A]_{(1,x,x^2,x^3)}^{(1,x,x^2)}B,$$I find that the two sides are not equal.

Where did I go wrong? Thanks in advance.

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haruspex
Problem. Show that$$[A]_{(1,x+1,(x+1)^2,(x+1)^3)}^{(1,x+1,(x+1)^2)} = C^{-1}[A]_{(1,x,x^2,x^3)}^{(1,x,x^2)}B.$$
that$$[A]_{(1,x,x^2,x^3)}^{(1,x,x^2)}= C^{-1}[A]_{(1,x+1,(x+1)^2,(x+1)^3)}^{(1,x+1,(x+1)^2)} B.$$