Exercise 12

Let $t\colon V\to V$ be a linear transformation and suppose $\vec{v}\in V$ is such that $t^{k}(\vec{v})=\vec{0}$ but $t^{k-1}(\vec{v})\neq\vec{0}$. Consider the $t$-string $\langle\vec{v},t(\vec{v}),\dots,t^{k-1}(\vec{v})\rangle$.

• 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.

• Prove that the restriction is nilpotent.

• Prove that the $t$-string is linearly independent and so is a basis for its span.

• Represent the restriction map with respect to the $t$-string basis.