We need the following definition for part (3) below. Vector spaces V and W are isomorphic if there is a linear transformation φ: V → W that is a bijection, in the sense that (i) φ(u) = φ(v) ⇒ u = v for all u, v∈V; and (ii) for all w ∈ W there exists v ∈ V such that φ(v)=w.Consider the set P2 := {f : R → R : f (x) = ax2 + bx + c for some a, b, c ∈ R}.So P2 is the set of polynomials of degree up to 2. This set is a vector space over R under addition given by(a1x2 +b1x+c1)+(a2x2 +b2x+c2) = (a1 +a2)x2 +(b1 +b2)x+(c1 +c2),and scalar multiplication given byλ(ax2 +bx+c) = λax2 + λbx + λc.Consider the function D : P2 → P2 given by D(ax2 + bx + c) = 2ax + b. a) Why did I call this function D? b) Prove that D is a linear transformation 1. c) Find an n such that P2 is isomorphic to Rn. (Don’t just state the value of n; prove why the vector spaces are isomorphic.) We need the following definition for part (c) below. Vector spaces V and W are isomorphicif there is a linear transformation o: V → W that is a bijection, in the sense that(i) ø(u) = $(v)» u = v for all u, v E V; and(ii) for all w E W there exists v e V such that o(v) := W.Consider the set P2 :={f:R → R: f(x) = ax² + bx + c for some a, b, c e R}. So P2 isthe set of polynomials of degree up to 2. This set is a vector space over R under additiongiven by(a1x² + b1x + c1) + (azx² + b2x + c2) = (a1 + az)x² + (b1 + b2)x + (c1 + c2),and scalar multiplication given byX(ax? + bx + c) = Aax² + Xbx + Xc.Consider the function D: P2 → P2 given by D(ax² + bx + c) = 2a.x + b.(a) Why did I call this function D?(b) Prove that D is a linear transformation'.(c) Find an n such that P2 is isomorphic to R". (Don't just state the value of n; provewhy the vector spaces are isomorphic.)

It is given that, D : P2 → P2 defined by D(ax2+bx+c) = 2ax+b Then, we have to answer the…

Answered: Consider the function D: P₂ → P2 given…

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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We need the following definition for part (3) below. Vector spaces V and W are isomorphic if there is a linear transformation φ: V → W that is a bijection, in the sense that

(i) φ(u) = φ(v) ⇒ u = v for all u, v∈V; and
(ii) for all w ∈ W there exists v ∈ V such that φ(v)=w.

Consider the set P2 := {f : R → R : f (x) = ax² + bx + c for some a, b, c ∈ R}.

So P₂ is the set of polynomials of degree up to 2. This set is a vector space over R under addition given by

(a₁x² +b₁x+c₁)+(a₂x² +b₂x+c₂) = (a₁ +a₂)x² +(b₁ +b₂)x+(c₁ +c₂),

and scalar multiplication given by

λ(ax² +bx+c) = λax² + λbx + λc.

Consider the function D : P2 → P2 given by D(ax² + bx + c) = 2ax + b.

a) Why did I call this function D?

b) Prove that D is a linear transformation 1.

c) Find an n such that P₂ is isomorphic to Rⁿ. (Don’t just state the value of n; prove why the vector spaces are isomorphic.)

We need the following definition for part (c) below. Vector spaces V and W are isomorphic
if there is a linear transformation o: V → W that is a bijection, in the sense that
(i) ø(u) = $(v)
» u = v for all u, v E V; and
(ii) for all w E W there exists v e V such that o(v) :
= W.
Consider the set P2 :=
{f:R → R: f(x) = ax² + bx + c for some a, b, c e R}. So P2 is
the set of polynomials of degree up to 2. This set is a vector space over R under addition
given by
(a1x² + b1x + c1) + (azx² + b2x + c2) = (a1 + az)x² + (b1 + b2)x + (c1 + c2),
and scalar multiplication given by
X(ax? + bx + c) = Aax² + Xbx + Xc.
Consider the function D: P2 → P2 given by D(ax² + bx + c) = 2a.x + b.
(a) Why did I call this function D?
(b) Prove that D is a linear transformation'.
(c) Find an n such that P2 is isomorphic to R". (Don't just state the value of n; prove
why the vector spaces are isomorphic.)

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