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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Question

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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