Define F,[x] = {f(x) € F[x]]deg ƒ (x) sn}. One can show F„[x] is a vector space. Let U = {f(x) € R,[x]||f(6) = 0}. a) Find a basis for U. Assume that for any field F, F,[x] is a vector space b) Extend the basis in part (a) to a basis of R4[x].
Define F,[x] = {f(x) € F[x]]deg ƒ (x) sn}. One can show F„[x] is a vector space. Let U = {f(x) € R,[x]||f(6) = 0}. a) Find a basis for U. Assume that for any field F, F,[x] is a vector space b) Extend the basis in part (a) to a basis of R4[x].
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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5) Define F,[x] = {f(x) € F[x]]degf(x) < n}: One can show F,[x] is a vector space.
Let U = {f(x) € R,[x]]f(6) = 0}.
a) Find a basis for U. Assume that for any field F, F,[x] is a vector space
b) Extend the basis in part (a) to a basis of R4[x].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa9774f45-b264-467e-8b06-716b402d428d%2Fadbe2d86-4a48-4120-b31b-478a1da09b3c%2Fefupu2h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:órigin.
5) Define F,[x] = {f(x) € F[x]]degf(x) < n}: One can show F,[x] is a vector space.
Let U = {f(x) € R,[x]]f(6) = 0}.
a) Find a basis for U. Assume that for any field F, F,[x] is a vector space
b) Extend the basis in part (a) to a basis of R4[x].
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In part (a.) wouldn't you have to prove that U = Span(B) as well as B being linearly independent to prove B is a basis for U?
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