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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) o(u) = 0(v) = u= v for all u, v € 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 € R}. So P, is the set of polynomials of degree up to 2. This set is a vector space over R under addition given by (a,a² + b,x + c1) + (aza² + bzx + c2) = (a1 + az)x² + (b + b2)x + (c + c2), and scalar multiplication given by A(ar² + bx + c) = \ax² + Abx + Ac. 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'. (c) Find an n such that Pa is isomorphic to R". (Don't just state the value of n; prove why the vector spaces are isomorphic.) (d) Let o : P2 → R" be the function from (c) you used to show that Pa and R" are isomorphic (where n is your value from (c)). Find a matrix Ap such that oo Doo- 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) o(u) = 0(v) = u= v for all u, v € 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 € R}. So P, is the set of polynomials of degree up to 2. This set is a vector space over R under addition given by (a,a² + b,x + c1) + (aza² + bzx + c2) = (a1 + az)x² + (b + b2)x + (c + c2), and scalar multiplication given by A(ar² + bx + c) = \ax² + Abx + Ac. 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'. (c) Find an n such that Pa is isomorphic to R". (Don't just state the value of n; prove why the vector spaces are isomorphic.) (d) Let o : P2 → R" be the function from (c) you used to show that Pa and R" are isomorphic (where n is your value from (c)). Find a matrix Ap such that oo Doo- is given by left multiplication by Ap- (e) Find the eigenvalues and eigenspaces of Ap.

Advanced Engineering Mathematics
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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part c-e please

1. We need the following definition for part (c) below. Vector spaces V and W are isomorphic if there is a linear transformation ¢: V → W that is a bijection, in the sense that (i) o(u) = 0(v) = u = v for all u, v € V; and (ii) for all w e W there exists v € V such that ø(v) = w. Consider the set P, := {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 (a,a² + b,x + c1) + (a,a² + b,¤ + c2) = (a1 + az)a² + (bì + b2)r + (c1 + c2), and scalar multiplication given by Max? + bx + c) = Aax? + Abx + dc. 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.) (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 ø o Doo-1 is given by left multiplication by Ap. (e) Find the eigenvalues and eigenspaces of Ap.
Transcribed Image Text:1. We need the following definition for part (c) below. Vector spaces V and W are isomorphic if there is a linear transformation ¢: V → W that is a bijection, in the sense that (i) o(u) = 0(v) = u = v for all u, v € V; and (ii) for all w e W there exists v € V such that ø(v) = w. Consider the set P, := {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 (a,a² + b,x + c1) + (a,a² + b,¤ + c2) = (a1 + az)a² + (bì + b2)r + (c1 + c2), and scalar multiplication given by Max? + bx + c) = Aax? + Abx + dc. 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.) (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 ø o Doo-1 is given by left multiplication by Ap. (e) Find the eigenvalues and eigenspaces of Ap.
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