A contour diagram (level-line plot) shows a linear functional on

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

A contour diagram (level-line plot) shows a linear functional ω\omega on R2\mathbb R^2, expressed in the standard basis x^,y^\hat{\mathbf x},\hat{\mathbf y}. The level line ω=0\omega=0 runs through the origin along the direction (2,−1)(2,-1), and the neighbouring level line ω=1\omega=1 passes through the point (1,0)(1,0); all the level lines are straight, parallel to (2,−1)(2,-1), and evenly spaced. ((a)) Read off the component row (ω1,ω2)(\omega_1,\omega_2) of ω\omega in the standard dual basis, so that ω(x,y)=ω1x+ω2y\omega(x,y)=\omega_1x+\omega_2y. ((b)) Evaluate ω(3,4)\omega(3,4).

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