In the context of linear algebra and functional analysis, a **cyclic subspace** is a specific type of subspace generated by the action of a linear operator on a particular vector. Often discussed in relation to operators on Hilbert spaces or finite-dimensional vector spaces, a cyclic subspace can be defined as follows: Let \( A \) be a linear operator on a vector space \( V \), and let \( v \in V \) be a vector.
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A cyclic subspace generated by under a linear operator is the span of its successive iterates . In finite dimension it is already generated by the first iterates. The first linear dependence expresses the next iterate in previous ones and proves invariance without using the Cayley-Hamilton theorem.