Answered: Let V=P(R) and for j≥1 define…

Let V=P(R) and for j≥1 define Tj(f(x))=f(j)(x), where f(j)(x) is the jth derivative of f(x). Prove that the set {T1,T2,…,Tn} is a linearly independent subset of L(V) for any positive integer n.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 69EQ: Let x=x(t) be a twice-differentiable function and consider the second order differential equation...

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Question

Let V=P(R) and for j≥1 define Tj(f(x))=f(j)(x), where f(j)(x) is the jth derivative of f(x). Prove that the set {T1,T2,…,Tn} is a linearly independent subset of L(V) for any positive integer n.

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