16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 6.7.9

Elementary Canonical Forms

Mathematics — Linear Algebra · even-odd-functions-invariant

Problem

Let VV be the vector space of continuous real-valued functions on the interval [−1,1][-1, 1] of the real line. Let WeW_e be the subspace of even functions, f(−x)=f(x)f(-x) = f(x), and let WoW_o be the subspace of odd functions, f(−x)=−f(x)f(-x) = -f(x). (a) Show that V=We⊕WoV = W_e \oplus W_o. (b) If TT is the indefinite integral operator (Tf)(x)=∫0xf(t) dt(Tf)(x) = \int_0^x f(t)\, dt are WeW_e and WoW_o invariant under TT?

Related theory

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