(Solution) - Consider the subspace W of D given by W span -(2025 Original AI-Free Solution)

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Consider the subspace W of D, given by
W = span (e2x, e-2x).
(a) Show that the differential operator D maps W into itself.
(b) Find the matrix of D with respect to B = {e2X, e-2x}.
(c) Compute the derivative of f(x) = e2x - 3e-zx indirectly, using Theorem 6.26, and verify that it agrees with f'(x) as computed directly.