On the interval define the inner product of two real functions by

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

On the interval [0,L][0,L] define the inner product of two real functions by ⟨f,g⟩=∫0Lf(x)g(x) dx\langle f,g\rangle = \int_0^L f(x)g(x)\,\mathrm{d}x. Take f(x)=cos⁡(πx/L)f(x)=\cos(\pi x/L) and g(x)=cos⁡(2πx/L)g(x)=\cos(2\pi x/L), which, unlike a pinned string's shapes, do not vanish at the ends. ((a)) Compute ⟨f,g⟩\langle f,g\rangle. ((b)) State whether ff and gg are orthogonal on [0,L][0,L].

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