Linear Algebra/Self-Composition

This subsection is optional, although it is necessary for later material in this section and in the next one.

A linear transformations $$t:V\to V$$, because it has the same domain and codomain, can be iterated. That is, compositions of $$t$$ with itself such as $$ t^2=t\circ t $$ and $$ t^3=t\circ t\circ t $$ are defined. Note that this power notation for the linear transformation functions dovetails with the notation that we've used earlier for their squared matrix representations because if $${\rm Rep}_{B,B}(t)=T$$ then $$ {\rm Rep}_{B,B}(t^j)=T^j $$.

These examples suggest that on iteration more and more zeros appear until there is a settling down. The next result makes this precise.

This graph illustrates Lemma 1.3. The horizontal axis gives the power $$j$$ of a transformation. The vertical axis gives the dimension of the rangespace of $$t^j$$ as the distance above zero&mdash; and thus also shows the dimension of the nullspace as the distance below the gray horizontal line, because the two add to the dimension $$n$$ of the domain. As sketched, on iteration the rank falls and with it the nullity grows until the two reach a steady state. This state must be reached by the $$n$$-th iterate. The steady state's distance above zero is the dimension of the generalized rangespace and its distance below $$n$$ is the dimension of the generalized nullspace.

Exercises
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