DE.07.4.Literacy

# DE.07.4.Literacy - L.3 Here is the flow of solutions of a...

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Mathematica Authors: Bruce Carpenter, Bill Davis and Jerry Uhl ©2001-2007 Publisher: Math Everywhere, Inc. Version 6.0 DE.07 Eigenvectors and Eigenvalues for Linear Systems LITERACY What you need to know when you're away from the machine. · L.1) a) Write down the matrix A that allows you to express the linear system x £ @ t D = 3.1 x @ t D - 6.0 y @ t D y £ @ t D = - 2.2 x @ t D + 1.7 y @ t D in the form 8 x £ @ t D , y £ @ t D< = A. 8 x @ t D , y @ t D< . b) Write down the matrix A that allows you to express the linear system x £ @ t D = - 2.9 y @ t D y £ @ t D = 1.6 x @ t D + 4.5 y @ t D in the form 8 x £ @ t D , y £ @ t D< = A. 8 x @ t D , y @ t D< . c) Given A = 1.4 - 2.5 4.1 0 , what do you get when you multiply out A. 8 x @ t D , y @ t D< ? · L.2) Here are six plots of trajectories for six different linear systems x £ @ t D = a x @ t D + b y @ t D y £ @ t D = c x @ t D + d y @ t D all with 8 x @ 0 D , y @ 0 D< = 8 5, 1 < : - 4 - 2 2 4 x - 3 - 2 - 1 1 2 3 y Plot 1 - 2 2 4 x - 2 - 1 1 2 3 4 y Plot 2 - 15 - 10 - 5 5 10 x - 15 - 10 - 5 5 10 y Plot 3 6 7 8 9 x 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 y Plot 4 - 20 - 10 10 20 x - 15 - 10 - 5 5 10 15 y Plot 5 1 2 3 4 5 x 0.4 0.6 0.8 1.0 y Plot 6 a) Use the visual evidence presented and make the calls: Which plot depicts a trajectory of a linear system whose coefficient matrix has eigenvalues: i) p + i q and p - i q with p > 0 and q = 0 ? ii) p + i q and p - i q with p < 0 and q ! 0 ? iii) p + i q and p - i q with p = 0 and q ! 0 ? iv) p and q with both p and q real and both p and q < 0? v) p and q with both p and q real at least one of p or q is positive? b) Here are the plots of solutions, y @ t D , of the same six systems all with the same starter data 8 x @ 0 D , y @ 0 D< = 8 5, 1 < : 1.0 1.5 2.0 2.5 3.0 t 2.0 2.5 3.0 3.5 4.0 4.5 y @ t D 5 10 15 20 t - 2 2 3 4 y @ t D 5 10 15 20 t 0.4 0.6 0.8 1.0 y @ t D 5 10 15 20 t - 3 3 y @ t D 5 10 15 20 t - 15 - 10 10 15 y @ t D 5 10 15 20 25 30 t - 15 - 10 10 15 y @ t D Your job is to pair the solution plots with the corresponding trajectory plots. · L.2) How do you tell that none of the following systems is a linear system? a) 8 x £ @ t D , y £ @ t D< = 8 Sin @ x @ t DD + 4.9 y @ t D , - 2.3 x @ t D + 0.4 y @ t D< , b) 8 x £ @ t D , y £ @ t D< = 9 - I y @ t D 2 - 1 M x @ t D - y @ t D , x @ t D= , c) 8 x £ @ t D , y £ @ t D< = 8 x @ t D y @ t D , - x @ t D< . · L.3) Here is the flow of solutions of a certain linear system x £ @ t D = - 1.75 x @ t D - 2.17 y @ t D y £ @ t D = - 1.8 x @ t D + 0.75 y @ t D : - 3 - 2 - 1 1 2 3 x - 4 - 2 2 4 y Pencil in the eigenvectors of the coefficient matrix A = - 1.75 - 2.17 - 1.8 0.75 . Explain how you see at a glance that one eigenvalue of the matrix A = - 1.75 - 2.17 - 1.8 0.75 is positive and the other is negative. Where are the straight line trajectories?

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DE.07.4.Literacy - L.3 Here is the flow of solutions of a...

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