What are eigenvalues and eigenvectors of polynomial vector space?

Question

Let $P_1$ be the vector space of the general polynomials of degree less than or equal to one and let $T: P_1 \rightarrow P_1$ be the linear transformation that takes the polynomial $1 + x$ to $5 + 2x$ and the polynomial $4 + x$ to $-2(4+x)$. What are the eigenvalues and eigenvectors?

Answer 1

HINT

1. $P_1$ in your example is like $\mathbb{R}^2$. Can you map your problem to vectors in $\mathbb{R}^2$?
2. After (1), the linear transformation can be represented by a matrix, for which you know how it maps 2 vectors. Can you construct the matrix from the given 2 mappings?
3. Diagonalize the matrix as usual
4. Eigenvalues stay the same, can you map eigenvectors back to $P_1$ using the inverse of the map you used in (1)?