Exercise I.B.8

Question

Let $\phi$ be a linear transformation given by
$$\phi:\mathbb{R}^2	o\mathbb{R}^2, \quad \begin{bmatrix} x \ y \end{bmatrix} \mapsto \begin{bmatrix} x+y \ 2x-y \end{bmatrix}$$
If the polynomial $p$ is defined heuristically by $p(x) = x^2-2x +1$, then what can $p(\phi) = p \circ \phi$ be?

Toğrul Topal · Accepted Answer

$
ewcommand{\Hom}[2]{\mathrm{Hom}\left( #1, #2 ight)}$
Since a polynomial can be defined over any structure with addition and multiplication, then we can assume that $p$ is defined over $(\Hom{\mathbb{R}^2}{\mathbb{R}^2},+,\circ)$ over $\mathbb{K}$ with $p(\phi)$ being
$$p(\phi) = \phi\circ \phi - 2 \phi + 1$$

Exercise I.B.8

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Exercise I.B.8

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