(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [itex]h: \mathbb{P_2} \rightarrow \mathbb{P_2}[/itex] represent the transformation [itex]h(p(x)) = xp'(x) + p(1-x)[/itex] for every polynomial [itex]p(x) \in \mathbb{P_2}[/itex]. Find the matrix of h with respect to the standard basis [itex]\{1, x, x^2\}[/itex]

2. Relevant equations

Matrix A of transformation: [itex]{\bf A} = [T(e_1) \hspace{0.5em} T(e_2) \hspace{0.5em} \ldots \hspace{0.5em} T(e_n)][/itex]

3. The attempt at a solution

Let [itex]p(x) = ax^2 + bx + c[/itex] then [itex]p'(x) = 2ax + b[/itex] and [itex]p(1-x) = a(1-x)^2 + b(1 - x) + c = ax^2 - 2ax + a + b - bx + c = ax^2 - 2ax - bx + a + b + c[/itex].

Now we can rewrite [itex]h(p(x)) = xp'(x) + p(1-x)[/itex] as

[itex]h(p(x)) = x(2ax + b) + ax^2 - 2ax - bx + a + b + c[/itex]

This is as far as I have gotten. I'm guessing that I am supposed to put [itex]h(p(1)) = x(2a + b) + a - 2a - b + a + b + c = 2ax + bx + c[/itex] making the first column of h's matrix [itex]\left[ \begin{array}{c} c \\ 2a + b \\ 0 \end{array} \right] [/itex]. Process is repeated for [itex]h(p(x))[/itex] and [itex]h(p(x^2))[/itex]

Am I on the right track?

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# Homework Help: Linear Algebra - Linear Transformation of a polynomial

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