Please do parts a, b, c 1. One application of linear transformations is to solving differential equations. Given a polynomialfe P2, we want to consider polynomials y € P2 satisfying the differential equation2y" + (-х + 3) y'+ 2у — fon R.To do so, we will consider the linear transformation T : P2 → P2 defined byТ\у) — 2y" + (-т + 3) у'+ 2уfor each y E P2.(a) Find the matrix of T with respect to the standard basis S = {1,x, x²} on P2.(b) The kernel of T is the set of all solutions y E P2 to the homogeneous differential equation2y" + (-а + 3) у'+ 2у 3D0on R.Find a basis and the dimension for the kernel of T.(c) Is T surjective? What does this tell you about the solution of the differential equation (*)?

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1. One application of linear transformations is to solving differential equations. Given a polynomial fe P2, we want to consider polynomials y € P2 satisfying the differential equation 2y" + (-х + 3) y'+ 2у — f on R. To do so, we will consider the linear transformation T : P2 → P2 defined by T(y) = 2y" + (-x + 3) y' + 2y for each y E P2. (a) Find the matrix of T with respect to the standard basis S = {1,x, x²} on P2. (b) The kernel of T is the set of all solutions y E P2 to the homogeneous differential equation 2y" + (-а + 3) у' + 2у — 0 on R. Find a basis and the dimension for the kernel of T. (c) Is T surjective? What does this tell you about the solution of the differential equation (*)?

1. One application of linear transformations is to solving differential equations. Given a polynomial fe P2, we want to consider polynomials y € P2 satisfying the differential equation 2y" + (-х + 3) y'+ 2у — f on R. To do so, we will consider the linear transformation T : P2 → P2 defined by T(y) = 2y" + (-x + 3) y' + 2y for each y E P2. (a) Find the matrix of T with respect to the standard basis S = {1,x, x²} on P2. (b) The kernel of T is the set of all solutions y E P2 to the homogeneous differential equation 2y" + (-а + 3) у' + 2у — 0 on R. Find a basis and the dimension for the kernel of T. (c) Is T surjective? What does this tell you about the solution of the differential equation (*)?

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 70EQ

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