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1) Let P2 = {polynomials f(x) = a2x²+a1x+ao, where ao, a1, a2 E R}, and define addition and scalar multiplication as follows: (a) Addition. Let f(x) = a2x² +a1x +ao and g(x) b2x² +b1x+ bo-. Then define (f + g)(x) = (a2 + b2)x² + (a1 + b1)x + (ao + bo) (b) Scalar multiplication. Let r E R, f(x) = a2x² + a1x + ao. Define (rf)(x) = (ra2)x² + (ra1)x + (rao) Show that P2 is a vector space.

1) Let P2 = {polynomials f(x) = a2x²+a1x+ao, where ao, a1, a2 E R}, and define addition and scalar multiplication as follows: (a) Addition. Let f(x) = a2x² +a1x +ao and g(x) b2x² +b1x+ bo-. Then define (f + g)(x) = (a2 + b2)x² + (a1 + b1)x + (ao + bo) (b) Scalar multiplication. Let r E R, f(x) = a2x² + a1x + ao. Define (rf)(x) = (ra2)x² + (ra1)x + (rao) Show that P2 is a vector space.

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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Determine addition and scalar multiplication as follows:

1) Let P2 = {polynomials f(x) = a2x²+a1x+a0, where ao, a1, a2 E R}, and define addition and scalar multiplication as follows: (a) Addition. Let f(x) = a2x²+a1x+ao and g(x) = b2x² +b1x+bo. Then define (f + g)(x) = (a2 + b2)x² + (a1 + b1 )æ + (ao + bo) (b) Scalar multiplication. Let r E R, f(x) = a2x² + a1x + ao. Define (rf)(x) = (ra2)x² + (ra1)x + (rao) Show that P2 is a vector space.
Transcribed Image Text:1) Let P2 = {polynomials f(x) = a2x²+a1x+a0, where ao, a1, a2 E R}, and define addition and scalar multiplication as follows: (a) Addition. Let f(x) = a2x²+a1x+ao and g(x) = b2x² +b1x+bo. Then define (f + g)(x) = (a2 + b2)x² + (a1 + b1 )æ + (ao + bo) (b) Scalar multiplication. Let r E R, f(x) = a2x² + a1x + ao. Define (rf)(x) = (ra2)x² + (ra1)x + (rao) Show that P2 is a vector space.
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