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.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.2: Determinants
Problem 13AEXP
Related questions
Question
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e02cf4e-ee42-49ab-81f6-cf259a222c3c%2Ff08bb7c5-4845-4846-bf80-9639dbc1ab74%2F80u4spc_processed.png&w=3840&q=75)
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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