(a) Under the assumption that u 1 , , u k form a Jordan chain for the coefficient matrix A, prove that the functions (10 17) are solutions to the system u Au (b) Prove that they are linearly independent u1(t) e w1, u (t) e(t w W), u (t) ext and, in general, u(t) e(tw tW2 W3), ti i 1jk (j i) W 23 (10 17)

Question: (a) Under the assumption that u 1 , . . . , u k form a Jordan chain for the coefficient matrix A, prove that

(a) Under the assumption that u₁, . . . , u_k form a Jordan chain for the coefficient matrix A, prove that the functions (10.17) are solutions to the system u̇ = Au.

u1(t)=e^w1, u (t) = e(t w + W), u; (t) = ext

(b) Prove that they are linearly independent.

u1(t)=e^w1, u (t) = e(t w + W), u; (t) = ext and, in general, u(t) = e(tw +tW2 + W3), ti-i 1jk. (j-i)! W 23 (10.17)

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