The complex solution space of has basis functions and

PHYSICS · Subjective · 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

The complex solution space of

\ddot{x}+\omega_0^2 x = 0

has basis functions

e^{i\omega_0 t}

and

e^{-i\omega_0 t}

, with

\omega_0

in rad s

^{-1}

and

t

in seconds. Using the complex inner product

\langle f,g\rangle = \int_0^T f^*(t)g(t)\,\mathrm{d}t

over one period

T=2\pi/\omega_0

, with

f(t)=e^{i\omega_0 t}

and

g(t)=e^{-i\omega_0 t}

, compute

\langle f,g\rangle

and state whether

f

and

g

are orthogonal over one period.

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