# Name for vectors in a Krylov space but not the preceding one

It seems to me that a useful concept to define when studying Krylov subspace methods is the idea of a vector $$v$$ 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., $$v$$ can be written as a polynomial $$v = p(A)b$$ of degree exactly $$n$$, with a nonzero leading coefficient.

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 word for it.

"Continuation vector" is something close, but it is too vague and specific to Arnoldi, when the concept appears in many other contexts.

• I have never seen a word for this. I need it when proving that Arnoldi's method works. I tend to shorten $K_j(A,b)$ to $K_j$ and just write $x \in K_{j+1}\setminus K_j$. Commented Dec 12, 2023 at 17:38
• It is possible that we need to cast a wider net. I am thinking of an infinite union of increasing sets as an onion. We are looking for the elements in the (j+1)\$st layer. This is not a phrase that have seen used, but it occured to me that there might be a good established turn of phrase outside of our very specialized context. Commented Dec 13, 2023 at 23:24