Consider 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

Consider the basis a1=(1,0,0)\mathbf a_1=(1,0,0), a2=(0,1,0)\mathbf a_2=(0,1,0), a3=(1,1,2)\mathbf a_3=(1,1,2) of R3\mathbb{R}^3 (Cartesian components in some fixed length unit; treat all numbers below as dimensionless multiples of that unit) and the metric gij=ai⋅ajg_{ij}=\mathbf a_i\cdot\mathbf a_j built from it. A linear functional ω∈V∗\omega\in V^* has covariant components ω1=2\omega_1=2, ω2=−1\omega_2=-1, ω3=3\omega_3=3 in this basis. ((a)) Using the reciprocal-basis machinery for a metric (the inverse matrix gijg^{ij}), compute the components ui=gijωju^i=g^{ij}\omega_j of the vector u\mathbf u representing ω\omega under the dot product, in the basis {a1,a2,a3}\{\mathbf a_1,\mathbf a_2,\mathbf a_3\}, and give the Cartesian components of u\mathbf u. ((b)) Let DD be the differentiation map on P4\mathcal P_4, the real polynomials of degree at most 44 in a real variable xx, Df=df/dxDf=\mathrm df/\mathrm dx. State briefly why P4\mathcal P_4 is closed under addition and under scalar multiplication, and give its dimension. Write the 5×55\times5 matrix of DD in the basis 1,x,x2,x3,x41,x,x^2,x^3,x^4. Using the rank-nullity theorem, find rank D\mathrm{rank}\,D, exhibit a basis for the null space N(D)\mathcal N(D), and exhibit a basis for the range R(D)\mathcal R(D) as an explicit finite-dimensional subspace of P4\mathcal P_4.

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