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.---------------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.)(d) Let φ: P2 →Rn be the function from (c) you used to show that P2 and Rn are isomorphic (where n is your value from (c)). Find a matrix AD such that φ ◦ D ◦ φ−1 is given by left multiplication by AD.(e) Find the eigenvalues and eigenspaces of AD . 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) = 6(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) = ax? + 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)+ (a2x² + b2x + c2) = (a1 + a2)x² + (b1 + b2)x + (c1+ c2),and scalar multiplication given byA(ax? + bx + c) = Xax² + Abx + dc.Consider the function D: P2 – P2 given by D(ax² + bx + c) = 2ax + b. (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.)(d) Let ø : P, → R" be the function from (c) you used to show that P, and R" areisomorphic (where n is your value from (c)). Find a matrix Ap such that oo Doo-1is given by left multiplication by Ap.(e) Find the eigenvalues and eigenspaces of Ap.

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Answered: (c) Find an n such that P₂ is…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 24EQ

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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.

---------------

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.)

(d) Let φ: P₂ →Rⁿ be the function from (c) you used to show that P₂ and Rⁿ are isomorphic (where n is your value from (c)). Find a matrix A_D such that φ ◦ D ◦ φ⁻¹ is given by left multiplication by A_D.

(e) Find the eigenvalues and eigenspaces of A_D

. 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) = 6(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) = ax? + 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)+ (a2x² + b2x + c2) = (a1 + a2)x² + (b1 + b2)x + (c1+ c2),
and scalar multiplication given by
A(ax? + bx + c) = Xax² + Abx + dc.
Consider the function D: P2 – P2 given by D(ax² + bx + c) = 2ax + b.

(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.)
(d) Let ø : P, → R" be the function from (c) you used to show that P, and R" are
isomorphic (where n is your value from (c)). Find a matrix Ap such that oo Doo-1
is given by left multiplication by Ap.
(e) Find the eigenvalues and eigenspaces of Ap.

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(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.)