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# Let be defined by . Show that T is not a linear operator on V

ISBN: 9780470432051 396

## Solution for problem 3 Chapter 8

Elementary Linear Algebra: Applications Version | 10th Edition

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Problem 3

Let be defined by . Show that T is not a linear operator on V.

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S343 Section 3.5 Notes- Nonhomogeneous Equations; Method of Undetermined Coefficients 10-13-16 ′′ ′  Nonhomogeneous equation- = + + = where ,, are given continuous functions on open interval o = 0 gives homogeneous equation + + = 0( )  Theorem 3.5.1- Suppose 1and a2e solutions of the nonhomogeneous ODE such that + + = = or 1 the2 − =2 1 = 0 is a solution to the homogeneous ODE o If 1,2} is also a fundamental set of solutions for the homogeneous equation, then 1 1 + 2 2= − 1 = 0 2( ) o Proof:  1,2satisfy 1 ( = and [2]( = ( )  Take difference of equations: 1]( − [ 2 ( = − = 0  Using linearity, 1 − 2= − 1 2]  So 1− 2]( = 0, which shows −1 is2solution to homogeneous ODE  Because all solutions can be expressed as linear combinations of fundamental sets of solutions (Theorem 3.2.4), we can write 1− 2 1 1 =2 2  Theorem 3.5.2- The general solution of the nonhomogeneous ODE can be written as =

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