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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) + (a2x² + b2x + c2) = (a1 + a2)x² + (b1 + b2)x + (c1 + c2), and scalar multiplication given by A(ax? + bx + c) = Xax² + Xbx + dc. Consider the function D: P2 → P2 given by D(ax² + bx + c) 2ах + b. (a) Why did I call this function D? (b) Prove that D is a linear transformation'.

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) + (a2x² + b2x + c2) = (a1 + a2)x² + (b1 + b2)x + (c1 + c2), and scalar multiplication given by A(ax? + bx + c) = Xax² + Xbx + dc. Consider the function D: P2 → P2 given by D(ax² + bx + c) 2ах + b. (a) Why did I call this function D? (b) Prove that D is a linear transformation'.

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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Consider the set P2 := {f:R → R : ƒ(x) the set of polynomials of degree up to 2. This set is a vector space over R under addition given by a.x? + bx + c for some a, b, c E R}. So P2 is (a1x2 + b1x + c1) + (a2x² + b2x + c2) = (a1 + a2)x² + (b1 + b2)x + (c1+c2), and scalar multiplication given by A(ax? + bx + c) = Xax² + Xbx + dc. 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'.
Transcribed Image Text:Consider the set P2 := {f:R → R : ƒ(x) the set of polynomials of degree up to 2. This set is a vector space over R under addition given by a.x? + bx + c for some a, b, c E R}. So P2 is (a1x2 + b1x + c1) + (a2x² + b2x + c2) = (a1 + a2)x² + (b1 + b2)x + (c1+c2), and scalar multiplication given by A(ax? + bx + c) = Xax² + Xbx + dc. 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'.
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