Given a linear transformation T: ℝn → ℝn, the iterates of T are the transformations given...

Given a linear transformation T: ℝn → ℝn, the iterates of T are the transformations given by

In general, the ith iterate of T is defined by Ti(x) = T(Ti−1(x)), i = 1, 2, …, where T0(x) = x.

(a) Find a formula for the ith iterate of T(x1, x2)=(x2, − x1).

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(b) Find a formula for the ith iterate of T(x1, x2, x3) = (x2, x3, 0).

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(c) Find a formula for Ti(x), where .