3y + x ·()-(²) = 2y . Let 0: R³ R³ be the linear mapping given by y (i) Write down the matrix A such that 0v = Av for all v € R³. (ii) Find the kernel of 0, write down a basis for it, and hence calculate the nullity of 0. (iii) Find the image of 0, write down a basis for it, and hence calculate the rank of 0. As a check, verify that the rank-nullity formula holds.

3y + x ·()-(²) = 2y . Let 0: R³ R³ be the linear mapping given by y (i) Write down the matrix A such that 0v = Av for all v € R³. (ii) Find the kernel of 0, write down a basis for it, and hence calculate the nullity of 0. (iii) Find the image of 0, write down a basis for it, and hence calculate the rank of 0. As a check, verify that the rank-nullity formula holds.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

3y + x
·0)-(²)
=
2y
1. Let 0 R³ R³ be the linear mapping given by y
(i) Write down the matrix A such that v = Av for all v € R³.
(ii) Find the kernel of 0, write down a basis for it, and hence calculate the nullity of 0.
(iii) Find the image of 0, write down a basis for it, and hence calculate the rank of 0.
As a check, verify that the rank-nullity formula holds.

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Step 2: Calculating matrix representation

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Step 3: Calculating kernel

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Step 4: Calculating basis of image of theta

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