Exercise 27 — Linear Algebra

Show that the range spaces and null spaces of powers of linear maps $t\colon V\to V$ form descending

V\supseteq\mathscr{R}(t)\supseteq\mathscr{R}(t^{2})\supseteq\ldots

and ascending

\text{$\{\hskip 1.00006pt\vec{0}\hskip 1.00006pt\}$}\subseteq\mathscr{N}(t)\subseteq\mathscr{N}(t^{2})\subseteq\ldots

chains. Also show that if $k$ is such that $\mathscr{R}(t^{k})=\mathscr{R}(t^{k+1})$ then all following range spaces are equal: $\mathscr{R}(t^{k})=\mathscr{R}(t^{k+1})=\mathscr{R}(t^{k+2})\ldots\,$ . Similarly, if $\mathscr{N}(t^{k})=\mathscr{N}(t^{k+1})$ then $\mathscr{N}(t^{k})=\mathscr{N}(t^{k+1})=\mathscr{N}(t^{k+2})=\ldots\,$ .

Hefferon, Linear Algebra, Maps Between Spaces, Range Space and Null Space, Exercise 27

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