A theorem in some notes I'm reading states:
Let $T \colon V \rightarrow V$ be linear and $E = (e_1,\dots,e_n)$ be a basis for V. Then E is a Jordan Basis for T and V if and only if there exists $k \geq 1$ and integers $i(0),\dots,i(k)$ such that $0 = i(0) < \dots < i(k) = n$ and
$(e_{i(t)+1},e_{i(t)+2}, \dots, e_{i(t+1)})$
Is a Jordan chain for all t with $0 \leq t < k$.
The conditions stated make no sense to me (it is probably a mistake in the notes), so what I'm looking for is an explanation of the last part of the theorem, or an exact statement of what it should be. Thanks
The result (which is more of a definition) is that E is a Jordan basis if and only if it is a basis which is a union of disjoint Jordan chains.