Question: I am preparing for GRE MATH SUBJECT TEST, I have reviewed many problems, some of these problems are not in GRE Math Practice books, but they are in some other books. I found the following problem, but not sure if it can be a GRE problem or no. And I do not even know how to start;

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> Let $P$ be a polynomial with integer coefficients satisfying $P(0)=1, P(1)=3, P'(0)=-1, P'(1)=10$. What is the minimum possible degree of $P$?
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> (A) $3$
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> (B) $4$
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> (C) $5$
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> (D) $6$
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> (E) No such $P$ exists.
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Answer: 
Yes, eigenvectors and eigenvalues are defined only for linear maps. And that map $f$ maps elements of $\Bbb R^2$ into elements of $\Bbb R^2$. In particular, it maps $(3,2)$ into $(4,3)$. Thinking about it as acting on vectors, by acting on their initial and final points, is misleading.
The question has already been answered in the straightforward way, but I just wanted to point out another perspective. There is a way to view this as a linear map in some sense. It is an affine map, which can be represented as a matrix by embedding it in $\mathbb{R}^3$ as follows.
$$ \begin{pmatrix} 1&0&0 \\ 1&1&0 \\ 1&0&1 \end{pmatrix} \begin{pmatrix} 1 \\ x \\ y \end{pmatrix} = \begin{pmatrix} 1 \\ x+1 \\ y+1 \end{pmatrix} $$
Thinking about the eigenvectors and eigenvalues of this matrix may be another instructive perspective on the original problem.