Let be the set of all real functions that satisfy the driven equation

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Mathematical Methods for Physics, Ch Vector spaces, bases, linear maps and dual vectors

Problem

Let SS be the set of all real functions x(t)x(t) that satisfy the driven equation x¨+ω02x=F0\ddot x + \omega_0^2 x = F_0, where ω0\omega_0 and F0F_0 are fixed nonzero constants, together with the usual addition of functions and multiplication by a real scalar. ((a)) Take two members x1(t),x2(t)∈Sx_1(t), x_2(t)\in S and show that their sum x1+x2x_1+x_2 does not belong to SS; state which forced equation the sum does satisfy. ((b)) Decide whether SS contains a zero element, i.e. a function x0(t)x_0(t) with x(t)+x0(t)=x(t)x(t)+x_0(t)=x(t) for every x∈Sx\in S. ((c)) Name the requirement or rule (closure, or one of the eight rules) that fails, and say whether choosing different fixed nonzero constants ω0,F0\omega_0, F_0 can repair the problem.

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