The transformation here is $T(f(x)) = f(x + 1) + f(x − 1)$ which is a linear endomorphism on V, where $V={f(x) ∈ R[x] : deg f(x) ≤ 2017}$
So I have to find the jcf J of T. Along with a basis of B.
$[T]_{B←B}= J$ for some ordered basis B of V.
Well for my work: with the transformation I went along and let $B=[ 1, x, . . . , x^{2017} ]$ which is just the standard basis of V.
Then I said $[T]_{B←B}= [[T(1)]_B [T(x)]_B . . . [T(x^{2017})]_B]$ which equals to the matrix
\begin{matrix} 2 & * & \cdots & * \\ & 2 & \cdots\\ & & \ddots& \\ &&&2 \end{matrix}
which is essentially just 2's across the diagonal and it is a 2018x2018 matrix
So J needs to equal to my matrix here. Can I say the JCF then is $J_{2,2018}$ The subscripts 2 is simply the eigenvalues (in this case all 2's on the diagonal) and the subscript 2018 is the number of times the eigenvalue repeats (this is the notation I learned at least.
I'm not sure if my work here makes sense however, I am really unsure with this one.