Question

Consider the linear system

Problem 1. (10 points) Consider the linear system 4 ' = [-12 -8 a. Find the eigenvalues and eigenvectors for the coefficient matrix. te and 12 = v2 = b. For each eigenpair in the previous part, form a solution of y' = Ay. Use t as the independent variable in your answers. gi(t) = and yz(t) = c. Does the set of solutions you found form a fundamental set (i.e., linearly independent set) of solutions? Choose

y⃗ ′=[6−124−8]y⃗ .

Answer 1

6-1 1-12 -8-1 * 9" F23] '=45 Let 't be the eigenvalue of 4. So, characterstic eqr. is: IA-d11=0 1 = (6-1) (-8-1) +48 = 0 12 + 2A 48 +48=0 2+2 d=0 7 d d + 2) = 0 d=0,- 2 Vi= [o] be the eigenvector for h=0 .. (A-O.I) V = 0 & A Vi=0 Now, [44][8][:] 6at ub=0 a =- ub a = 6 3 Let b=3, a = -2 for dia O Let V₂² [3] is eigenvector for [ o ] FR][-]-[:] the be eigenvector for [2=-2 (A +2 I) V₂ = 0 m=-20 82 tum=0 het l=1, m = -2 ( ملا V₂ = for de=-2 d,= o [3] is eigenvector [3] , d = (-2), v estry (:e=1) Vi= V2 = -2 Vi= edit pot V, - [ 4 V2 = edizt v₂ = e-2t [%] ezt 1-2e-2t Ang (c) Does the set of solution we find form fundamental Set. Yes estry