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Let T: C[0, 1] → C[0, 1] be defined by Tf(x) = f f(t)dt. (a) Show that T is linear. (b) Find T-¹: R(T) → C[0, 1], the inverse of T. (c) Determine if T-¹ is linear and bounded? (Hint: Consider function of polynomials degree n.) (d) Determine Ker(T-¹).

Let T: C[0, 1] → C[0, 1] be defined by Tf(x) = f f(t)dt. (a) Show that T is linear. (b) Find T-¹: R(T) → C[0, 1], the inverse of T. (c) Determine if T-¹ is linear and bounded? (Hint: Consider function of polynomials degree n.) (d) Determine Ker(T-¹).

Linear Algebra: A Modern Introduction
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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Let T: C[0, 1] → C[0, 1] be defined by = [² f Tf(x) = f(t)dt. (a) Show that T is linear. (b) Find T-¹: R(T) → C[0, 1], the inverse of T. (c) Determine if T-¹ is linear and bounded? (Hint: Consider function of polynomials degree n.) (d) Determine Ker(T-¹).
Transcribed Image Text:Let T: C[0, 1] → C[0, 1] be defined by = [² f Tf(x) = f(t)dt. (a) Show that T is linear. (b) Find T-¹: R(T) → C[0, 1], the inverse of T. (c) Determine if T-¹ is linear and bounded? (Hint: Consider function of polynomials degree n.) (d) Determine Ker(T-¹).
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