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Stability and subspaces of a linear system

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Please use a step-by-step method to solve the two following exercises:

Exercise 1:

Consider the linear system x' = Ax, where A is a 2 x 2 matrix with lamda in the diagonal as follows;

A = [lamda, -2]
[1 , lamda],

and lamda is real. Determine if the system has a saddle, node, focus, or center at the origin and determine the stability of each node or focus.

Exercise 2:

Consider the linear system x' = Ax, where A is a 3x3 matrix as follows:

A = [ 1, 0 , 0 ]
[ 0, -1, -1 ]
[ 0 , 1, -1 ]

a) Find the general solution of the linear system.

b) Find the stable, unstable, and the center subspaces E^s, E^u and E^c.

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Solution Summary

We solve problems involving the stability and subspaces of dynamic linear systems.

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