Let be a real vector space with the dot product

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

Conditions of Validity, Ch Vector spaces, linear maps and the dual: the Riesz theorem

Problem

Let VV be a real vector space with the dot product, basis {ei}\{\mathbf{e}_i\}, metric gij=ei⋅ejg_{ij}=\mathbf{e}_i\cdot\mathbf{e}_j, and let a functional ω∈V∗\omega\in V^* have components ωi=ω(ei)\omega_i=\omega(\mathbf{e}_i). The Riesz representer u\mathbf{u} of ω\omega has components uiu^i found from the Gram system ∑iuigij=ωj\sum_i u^i g_{ij}=\omega_j. A new basis is introduced by ej′=Sijei\mathbf{e}'_j=S^i{}_j\mathbf{e}_i, with S\mathsf{S} real and invertible. ((a)) Show that the metric transforms as g′=STgS\mathsf{g}'=\mathsf{S}^{\mathrm{T}}\mathsf{g}\mathsf{S} and that the functional's components transform as ω′=STω\omega'=\mathsf{S}^{\mathrm{T}}\omega. ((b)) Show that the representer's components in the new basis are u′=S−1uu'=\mathsf{S}^{-1}u, i.e.\ that u\mathbf{u} transforms with the same rule as an ordinary vector, and say why this had to be true without further calculation.

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