Let \(\mathcal{S}\) be the linear space of infinite sequences of real numbers \(x:=\left(x_{1}, x_{2}, \ldots\right) .\) Define the linear map \(L: \mathcal{S} \rightarrow \mathcal{S}\) by

\[

L x:=\left(x_{1}+x_{2}, x_{2}+x_{3}, x_{3}+x_{4}, \ldots\right) .

\]

a) Find a basis for the nullspace of \(L\). What is its dimension?

b) What is the image of \(L ?\) Justify your assertion.

c) Compute the eigenvalues of \(L\) and an eigenvector corresponding to each eigenvalue.