1e please 1. We need the following definition for part (c) below. Vector spaces V and W are isomorphicif there is a linear transformation ø: V → W that is a bijection, in the sense that(i) Ó(u) = ¢(v)= u = v for all u, v e V; and(ii) for all w e W there exists v € V such that $(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(a12? + b1x + c1) + (a2a² + b2x + c2) = (a1 + a2)x² + (b1 + b2)x + (c1 + c2),and scalar multiplication given byA(ax² + bx + c) = \ax² + Xbx + Ac.Consider the function D: P2 → P2 given by D(ax? + bx + c) = 2ax +b.(a) Why did I call this function D?(b) Prove that D is a linear transformation'.(c) Find an n such that P, is isomorphic to R". (Don't just state the value of n; provewhy the vector spaces are isomorphic.)(d) Let ø : P2 → R" be the function from (c) you used to show that P2 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.

Given that D:P2→P2 defined as Dax2+bx+c=0·x2+2ax+b. Here P2 stands for vector space of polynomials…

1. 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) (u) = 6(v) = u = v for all u, v e V; and (ii) for all w eW there exists v E V such that ¢(v) = w. Consider the set P2 := {f: R →R: f (x) = a.x² + ba + 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 (a122 + bịa + ci) + (a2a² + b2x + c2) = (a1 + a2)a² + (bi + b2)x + (cı + c2), and scalar multiplication given by (ax? + bx + c) = Aax? + Xbx + c. Consider the function D: P2 P2 given by D(ax? + bx + c) = 2ax + 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.) (d) Let ø : P2 → R" be the function from (c) you used to show that P2 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.

1. 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) (u) = 6(v) = u = v for all u, v e V; and (ii) for all w eW there exists v E V such that ¢(v) = w. Consider the set P2 := {f: R →R: f (x) = a.x² + ba + 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 (a122 + bịa + ci) + (a2a² + b2x + c2) = (a1 + a2)a² + (bi + b2)x + (cı + c2), and scalar multiplication given by (ax? + bx + c) = Aax? + Xbx + c. Consider the function D: P2 P2 given by D(ax? + bx + c) = 2ax + 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.) (d) Let ø : P2 → R" be the function from (c) you used to show that P2 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.

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

1e please

1. We need the following definition for part (c) below. Vector spaces V and W are isomorphic
if there is a linear transformation ø: V → W that is a bijection, in the sense that
(i) Ó(u) = ¢(v)
= u = v for all u, v e V; and
(ii) for all w e W there exists v € V such that $(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
(a12? + b1x + c1) + (a2a² + b2x + c2) = (a1 + a2)x² + (b1 + b2)x + (c1 + c2),
and scalar multiplication given by
A(ax² + bx + c) = \ax² + Xbx + Ac.
Consider the function D: P2 → P2 given by D(ax? + bx + c) = 2ax +b.
(a) Why did I call this function D?
(b) Prove that D is a linear transformation'.
(c) Find an n such that P, is isomorphic to R". (Don't just state the value of n; prove
why the vector spaces are isomorphic.)
(d) Let ø : P2 → R" be the function from (c) you used to show that P2 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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