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7. Let V be a vector space and l: V → R be a linear map. If z E V is not in the nullspace of l, show that every r E V can be decomposed uniquely as r = v+ cz, where v E V is in the nullspace of l and cER is a scalar.

7. Let V be a vector space and l: V → R be a linear map. If z E V is not in the nullspace of l, show that every r E V can be decomposed uniquely as r = v+ cz, where v E V is in the nullspace of l and cER is a scalar.

Advanced Engineering Mathematics
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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solve the question 7 with complete explanation asap
6. Consider the sphere a? + y? + (z - 3)2 = 1. Find the point P (a, b, c) on the sphere such that the tangent plane to the sphere at P intersects the ry-plane in the line y = 3x. 7. Let V be a vector space and l: V →R be a linear map. If z E V is not in the nullspace of €, show that every r EV can be decomposed uniquely as r = v+cz , where v E V is in the nullspace of l and c ER is a scalar. 8. Consider the three functions a(r), b(x), and c(r) given by: 3n 3n+1 a(x) = E (3n)! b(1x) = E; 3n+2 -Σ c(x) (Зп + 1)! (Зп + 2)! n=0 n=0 n=0 (a) Determine the radius of convergence for each of function. (b) Give a formula for the nth derivative a(n) (r) for all n E N.
Transcribed Image Text:6. Consider the sphere a? + y? + (z - 3)2 = 1. Find the point P (a, b, c) on the sphere such that the tangent plane to the sphere at P intersects the ry-plane in the line y = 3x. 7. Let V be a vector space and l: V →R be a linear map. If z E V is not in the nullspace of €, show that every r EV can be decomposed uniquely as r = v+cz , where v E V is in the nullspace of l and c ER is a scalar. 8. Consider the three functions a(r), b(x), and c(r) given by: 3n 3n+1 a(x) = E (3n)! b(1x) = E; 3n+2 -Σ c(x) (Зп + 1)! (Зп + 2)! n=0 n=0 n=0 (a) Determine the radius of convergence for each of function. (b) Give a formula for the nth derivative a(n) (r) for all n E N.
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