Let V = P5(Q) the vector space consisting of polynomials in the variable x of degree at most 5. Let w E L(V) be a linear map defined by o(f) = x²f" – 6xf' + 12f for any f E V. (a) Prove that p is a linear transformation. T:V → W is a linear transformation if the following conditions hold: 1) T(x + у) %3D T(х) + T(),х, у € V. 2) T(ax) = aT(x); x E V & a scalar. Pf: 1) p(f + g) = x²f + g)" – 6x(f + g)' + 12(f + g) = x²(f" + g") – 6x(f' + g') + 12(f + g) = (x²f" – 6xf' + 12f) + (x²g" – 6xg' + 12g) = p(f) + «(g) d? (f + g) : d²f) + d?(g)\ dx2 dx? dx2 2) p(af) = x²(af)" – 6x(af)' + 12(af) = x²(af") – 6x(af") + 12(af) = ax²f" – a6xf' + a12f = ap(f) ) φ(αf) - αφ f) %3D Hence, p is a linear transformation. (b) Determine a basis for the image and nullspace of p.
Let V = P5(Q) the vector space consisting of polynomials in the variable x of degree at most 5. Let w E L(V) be a linear map defined by o(f) = x²f" – 6xf' + 12f for any f E V. (a) Prove that p is a linear transformation. T:V → W is a linear transformation if the following conditions hold: 1) T(x + у) %3D T(х) + T(),х, у € V. 2) T(ax) = aT(x); x E V & a scalar. Pf: 1) p(f + g) = x²f + g)" – 6x(f + g)' + 12(f + g) = x²(f" + g") – 6x(f' + g') + 12(f + g) = (x²f" – 6xf' + 12f) + (x²g" – 6xg' + 12g) = p(f) + «(g) d? (f + g) : d²f) + d?(g)\ dx2 dx? dx2 2) p(af) = x²(af)" – 6x(af)' + 12(af) = x²(af") – 6x(af") + 12(af) = ax²f" – a6xf' + a12f = ap(f) ) φ(αf) - αφ f) %3D Hence, p is a linear transformation. (b) Determine a basis for the image and nullspace of p.
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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I need detailed help on how find basis and nullspace of this problem 2b (higfhlighted), thanyou so much.
Transcribed Image Text:2) Let V = P3 (Q) the vector space consisting of polynomials in the variable x of degree at most 5.
Let p E L(V) be a linear map defined by o(f) = x²f" – 6xf' + 12f for any f e V.
%3D
(a) Prove that
is a linear transformation.
T:V → W is a linear transformation if the following conditions hold:
1) T(x + y) = T(x) + T(y),x, y E V.
2) T (ах) — аТ (х);х € V &a scalar.
Pf:
1) p(f + g) = x²(f + g)" – 6x(f +g)' + 12(f + g)
= x²(f" + g") – 6x(f' + g') + 12(f + g)
= (x²f" – 6xf' + 12f) + (x²g" – 6xg' + 12g)
%3D
d²f)
= p(f)+ ¢(g)
..(: f + g) = d²M+ d°@)
dx²
dx²
dx²
2) p(af) = x²(af)" – 6x(af)' + 12(af)
= x² (af") – 6x(af') + 12(af)
= ax²f" – a6xf' + a12f
= aw(f)
:) φ(αf)- αφf)
Hence, p is a linear transformation.
(b) Determine a basis for the image and nullspace of p.
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