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Exercise 5.A.32

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Proof. Let $D$ denote the differentiation operator on the vector space of real-valued functions on $ℝ$ . Note that

D e^{λ_{j} x} = λ_{j} e^{λ_{j} x}

for each $j = 1, \dots, n$ . Then $e^{λ_{1} x}, \dots, e^{λ_{n} x}$ are eigenvectors corresponding to distinct eigenvalues. 5.10 now implies that they are linearly independent. □

celiopassos

2017-10-06 00:00

Exercise 5.A.32

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