Let t: V → V be a linear transformation and suppose →v ∈ V is such that tk(→v) = →0 but tk-1(→v) ↔ →0. Consider the t-string h→v, t(→v),....., tk-1(→v)i.
(a) Prove that t is a transformation on the span of the set of vectors in the string, that is, prove that t restricted to the span has a range that is a subset of the span. We say that the span is a t-invariant subspace.
(b) Prove that the restriction is nilpotent.
(c) Prove that the t-string is linearly independent and so is a basis for its span.
(d) Represent the restriction map with respect to the t-string basis.