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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Transcribed Image Text: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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Show linear dependence in part (ii) to show that ({1,0,1}) is a basis.
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