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It seems to me that a useful concept to define when studying Krylov subspace methods is the idea of a vector that belongs to a Krylov subspace $\mathcal{K}_{n+1}(A,b)$ but not to the preceding one $\mathcal{K}_{n}(A,b)$, i.e., they can be written as a polynomial $p(A)b$ of degree exactly $n$, with nonsingular leading term.

Is there a standard name or notation for vectors that belong to $\mathcal{K}_{n+1}(A,b) \setminus \mathcal{K}_{n}(A,b)$? Or, how can I describe or denote them? Something like "$v$ is a proper vector in $\mathcal{K}_{n+1}(A,b)$" sounds good, but I don't want to invent a new term when there already exists a perfectly cromulent word for it1.

1: sorry, I had to.

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