LetV = R[x]3, W = R[x]6be the vector spaces consisting of zero and polynomials in x of degree ≤ 3 and degree ≤ 6 respectively.This question concerns the map α : V → W which sends a polynomial f(x) in V to f(x^2) in W.(a) Show that α is linear as a map V →W.(b) Find the nullity ν(α) and the rank ρ(α).(c) Let U = Image(α) ⊆ W and find a subspace Z ⊆ W such that W = U ⊕ Z. LetV = R[x] 3, W = R[x]6be the vector spaces consisting of zero and polynomials in x of degree ≤ 3 and degree< 6 respectively.This question concerns the map a : V → W which sends a polynomial f(x) in V tof(x²) in W.(a) Show that a is linear as a map V → W.(b) Find the nullity v(a) and the rank p(a).(c) Let U = Image(a) CW and find a subspace ZC W such that W = UZ.Justify your answers.

Answered: Image /qna-images/answer/ee1eac55-9145-48c0-b572-14a9c8ea3161.jpg

V = R[x] 3, W = R[x]6 be the vector spaces consisting of zero and polynomials in x of degree ≤ 3 and degree 6 respectively. This question concerns the map :: V→ W which sends a polynomial f(x) in V to f(x²) in W. (a) Show that a is linear as a map V → W. (b) Find the nullity v(a) and the rank p(a). (c) Let U = Image(a) C W and find a subspace ZC W such that W = U Z.

V = R[x] 3, W = R[x]6 be the vector spaces consisting of zero and polynomials in x of degree ≤ 3 and degree 6 respectively. This question concerns the map :: V→ W which sends a polynomial f(x) in V to f(x²) in W. (a) Show that a is linear as a map V → W. (b) Find the nullity v(a) and the rank p(a). (c) Let U = Image(a) C W and find a subspace ZC W such that W = U Z.

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

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Let
V = R[x]3, W = R[x]6

be the vector spaces consisting of zero and polynomials in x of degree ≤ 3 and degree ≤ 6 respectively.

This question concerns the map α : V → W which sends a polynomial f(x) in V to f(x^2) in W.

(a) Show that α is linear as a map V →W.

(b) Find the nullity ν(α) and the rank ρ(α).

Let
V = R[x] 3, W = R[x]6
be the vector spaces consisting of zero and polynomials in x of degree ≤ 3 and degree
< 6 respectively.
This question concerns the map a : V → W which sends a polynomial f(x) in V to
f(x²) in W.
(a) Show that a is linear as a map V → W.
(b) Find the nullity v(a) and the rank p(a).
(c) Let U = Image(a) CW and find a subspace ZC W such that W = UZ.
Justify your answers.

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