Two ideally-mixed tanks hold a dye of concentrations (in )

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Mathematical Methods for Physics, Ch Vector spaces, bases, linear maps and dual vectors

Problem

Two ideally-mixed tanks hold a dye of concentrations c1,c2c_1, c_2 (in mol m−3\mathrm{mol\,m^{-3}}), collected as c=(c1,c2)\mathbf{c} = (c_1, c_2). First-order flow between and out of the tanks gives c˙=Mc\dot{\mathbf{c}} = \mathsf{M}\mathbf{c} with the linear map M=(−312−2)s−1\mathsf{M} = \begin{pmatrix}-3 & 1\\ 2 & -2\end{pmatrix}\mathrm{s^{-1}} in the standard basis {(1,0),(0,1)}\{(1,0),(0,1)\}. A detector reports the signal S=w⋅cS = \mathbf{w}\cdot\mathbf{c} with weights w=(1,3)\mathbf{w} = (1, 3) (dimensionless), a dual vector on the concentration vector c\mathbf{c}. ((a)) Find the two eigenvalues λ1>λ2\lambda_1 > \lambda_2 of M\mathsf{M} and an eigenvector for each, scaled so that its first component is 11; call these v1\mathbf{v}_1 and v2\mathbf{v}_2. ((b)) Using v1,v2\mathbf{v}_1, \mathbf{v}_2 as the columns of a change-of-basis matrix P\mathsf{P}, compute M′=P−1MP\mathsf{M}' = \mathsf{P}^{-1}\mathsf{M}\mathsf{P} and confirm it is diagonal with entries λ1,λ2\lambda_1, \lambda_2. ((c)) Confirm tr M=tr M′\mathrm{tr}\,\mathsf{M} = \mathrm{tr}\,\mathsf{M}' and det⁡M=det⁡M′\det\mathsf{M} = \det\mathsf{M}'. ((d)) At an instant when c=(4.0,2.0) mol m−3\mathbf{c} = (4.0, 2.0)\,\mathrm{mol\,m^{-3}} (standard-basis components), compute the signal S=w⋅cS = \mathbf{w}\cdot\mathbf{c}. ((e)) Find the components c′=P−1c\mathbf{c}' = \mathsf{P}^{-1}\mathbf{c} of the same state in the eigenvector basis. Then find the weights w′\mathbf{w}' for which w′⋅c′\mathbf{w}'\cdot\mathbf{c}' gives the same signal as w⋅c\mathbf{w}\cdot\mathbf{c} for every state, and confirm that S′=w′⋅c′S' = \mathbf{w}'\cdot\mathbf{c}' equals SS from part (d). ((f)) State which quantities from parts (a)-(c) are the same in both bases, and explain why w\mathbf{w} could not be transformed by the rule P−1\mathsf{P}^{-1} that c\mathbf{c} was.

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