The Cyclic Subspace
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Mathematics/Linear Algebra
The Cyclic SubspaceDefinition 1. Let $T \in \mathcal{L}(V)$, and let a nonzero vector $x \in V$. The subspace $W = \langle x, T(x), T^2(x), ... \rangle$ is called the $T$-cyclic subspace of $V$ generated by $x$.Theorem 1Theorem 1. Let $T \in \mathcal{L}(V)$, and let $W$ be the $T$-cyclic subspace of $V$ generated by $\mathbf{0} \neq x \in V$. Then(a) $W$ is $T$-invariant.(b) Any $T$-invariant su..