In a laboratory a displacement is recorded in the basis

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

In a laboratory a displacement is recorded in the basis {e1,e2}\{\mathbf e_1,\mathbf e_2\}, each basis vector representing 1 m1\ \mathrm m along a fixed direction. A recorded displacement has components vi=(2,5)v^i=(2,5), pure numbers counting basis vectors, so the actual displacement is 2e1+5e22\mathbf e_1+5\mathbf e_2. A linear functional ω\omega, representing an elastic energy release rate, has components ωi=ω(ei)=(4,−2)\omega_i=\omega(\mathbf e_i)=(4,-2), the joules released per one-metre basis step, so that ω(v)=ω1v1+ω2v2\omega(\mathbf v)=\omega_1v^1+\omega_2v^2 is in joules. ((a)) The laboratory adopts a coarser basis ei′=4ei\mathbf e'_i=4\mathbf e_i (each new basis vector is 4 m4\ \mathrm m long), so S=4I\mathsf S=4\mathsf I. Find the new components v′iv'^i of the same displacement and the new components ωi′\omega'_i of the same functional. ((b)) Evaluate ω(v)\omega(\mathbf v) using both the primed and unprimed components and confirm they agree.

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