The temperature along a thin uniform rod of length (in metres) is modelled

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

The temperature along a thin uniform rod of length LL (in metres) is modelled, over a short stretch, by a polynomial of degree at most 1, T(x)=a+bx,x∈[0,L],T(x) = a + bx, \qquad x\in[0,L], with aa in kelvin and bb in kelvin per metre. ((a)) Writing P1\mathcal P_1 for the set of all such polynomials with the usual addition and scalar multiplication of functions, confirm that P1\mathcal P_1 is closed under both operations, so that it is a vector space, and state a basis {e1,e2}\{e_1,e_2\} for it. ((b)) Construct the dual basis {ω1,ω2}\{\omega^1,\omega^2\} of P1∗\mathcal P_1^* for the basis {1,x}\{1,x\}, i.e. the linear functionals with ωi(ej)=δji\omega^i(e_j)=\delta^i_j. ((c)) A thermocouple at the far end of the rod reads the functional ϕ(T)=T(L)\phi(T)=T(L). Find the numbers ϕ1,ϕ2\phi_1,\phi_2 such that ϕ=ϕ1ω1+ϕ2ω2\phi=\phi_1\omega^1+\phi_2\omega^2.

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