Let carry the standard 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 V=R2V=\mathbb{R}^2 carry the standard dot product, and let e1=(1,0)\mathbf e_1=(1,0), e2=(1,1)\mathbf e_2=(1,1) be a basis (all components here are dimensionless real numbers). A linear functional ω∈V∗\omega\in V^* has components ω1=ω(e1)=3\omega_1=\omega(\mathbf e_1)=3, ω2=ω(e2)=5\omega_2=\omega(\mathbf e_2)=5. ((a)) For a general basis e1,…,en\mathbf e_1,\ldots,\mathbf e_n of a real inner-product space, with Gram matrix gij=ei⋅ejg_{ij}=\mathbf e_i\cdot\mathbf e_j, derive the linear condition that the components uku^k of a vector u=ukek\mathbf u=u^k\mathbf e_k must satisfy for ω(v)=u⋅v\omega(\mathbf v)=\mathbf u\cdot\mathbf v to hold for every v∈V\mathbf v\in V. Show that gijg_{ij} is invertible and hence that exactly one such u\mathbf u exists. Apply this to the numbers above to find (u1,u2)(u^1,u^2) in the basis {e1,e2}\{\mathbf e_1,\mathbf e_2\}, and the Cartesian components of u\mathbf u. ((b)) Change basis to e1′=e1\mathbf e'_1=\mathbf e_1, e2′=e1+e2\mathbf e'_2=\mathbf e_1+\mathbf e_2, so that ej′=Sijei\mathbf e'_j=S^i{}_j\mathbf e_i with S=(1101)S=\begin{pmatrix}1&1\\0&1\end{pmatrix}. Derive the rule by which the components of ω\omega change under this substitution and compute ω1′,ω2′\omega'_1,\omega'_2. Recompute the Gram matrix gjk′=ej′⋅ek′g'_{jk}=\mathbf e'_j\cdot\mathbf e'_k and the representer's components (u′1,u′2)(u'^1,u'^2) in the new basis, and confirm that u′1e1′+u′2e2′u'^1\mathbf e'_1+u'^2\mathbf e'_2 reproduces the same vector u\mathbf u found in part (a).

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