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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Let
V = R[x]3, W = R[x]6
be the
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.
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F16556e8d-f4c5-4c0e-af47-6e925b310d10%2Fee1eac55-9145-48c0-b572-14a9c8ea3161%2F4m5872_processed.png&w=3840&q=75)
Transcribed Image Text: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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