Recall that the quadratic polynomial f(t) = at² +bt+c, with a, b, c € R, is ≥ 0 for all t or ≤ 0 for all t if and only if it has either no real roots or one (double) real root. From the quadratic formula, this happens if and only if b2 - 4ac ≤ 0. Define f(t) = (X + tY, X + tY), where t is a real variable. Explain why f(t) ≥ 0 for all t. Using the algebraic properties of the inner product, write f(t) as a quadratic polynomial in with coefficients a = (Y, Y), b = 2(X, Y), and c = = (X, X). Using the last sentence of part (a), prove |(X, Y)| ≤ ||X|| ||Y||.
Recall that the quadratic polynomial f(t) = at² +bt+c, with a, b, c € R, is ≥ 0 for all t or ≤ 0 for all t if and only if it has either no real roots or one (double) real root. From the quadratic formula, this happens if and only if b2 - 4ac ≤ 0. Define f(t) = (X + tY, X + tY), where t is a real variable. Explain why f(t) ≥ 0 for all t. Using the algebraic properties of the inner product, write f(t) as a quadratic polynomial in with coefficients a = (Y, Y), b = 2(X, Y), and c = = (X, X). Using the last sentence of part (a), prove |(X, Y)| ≤ ||X|| ||Y||.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.5: Systems Of Linear Equations In More Than Two Variables
Problem 43E
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Transcribed Image Text:9. High school algebra and Cauchy-Schwarz:
(a) Recall that the quadratic polynomial ƒ(t) = at² + bt+c, with a, b, c =
R, is ≥ 0 for all t or ≤ 0 for all t if and only if it has either no
real roots or one (double) real root. From the quadratic formula, this
happens if and only if b² - 4ac ≤ 0.
(b) Define f(t) = (X + tY, X + tY), where t is a real variable. Explain
why f(t) ≥ 0 for all t.
(c) Using the algebraic properties of the inner product, write f(t) as a
quadratic polynomial in t with coefficients a = (Y, Y), b = 2(X,Y),
= (X, X).
and c =
(d) Using the last sentence of part (a), prove |(X, Y)| ≤ ||X|| ||Y||.
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