Polynomial/Bounded degree/Value/Affine space/Exercise

Let ${}d\in \mathbb {N} _{+}$, and let ${}K$ denote a field. Let ${}n$ different elements ${}a_{1},\ldots ,a_{n}\in K$ and ${}n$ elements ${}b_{1},\ldots ,b_{n}\in K$ be given. Show that the set ${}E$ of all polynomials ${}P$ of degree at most ${}d$, satisfying

${}P(a_{i})=b_{i}\,$

for ${}i=1,\ldots ,n$, is an affine subspace of ${}K[X]_{\leq d}$. What is the corresponding linear subspace? What can we say about the dimension of ${}E$, when is ${}E$ empty?

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