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(b) Let a = (a1, a2,..., an) and b = (b1, b2, ..., bn) be any two vectors in R". The inner product (dot product) of these two vectors are defined as a ·b = a,b1 + a,b2 + + anbn, and also the norms of these vectors are defined as Vā a = Važ + až + + a유, |||| = V5 .5 Vb+ b을 + + b. Prove the Cauchy-Schwarz inequality (a · b)? < |lä||2||b||°, that is the inequality (abı + azb2 + + anbn)² < (a² + a² + · + a) (bỉ + b3+. + b). Hint: Consider the function f(x) = (a1x + b1)² + (a2x + b2)² + . .. + (anx + br)² and apply (a).

(b) Let a = (a1, a2,..., an) and b = (b1, b2, ..., bn) be any two vectors in R". The inner product (dot product) of these two vectors are defined as a ·b = a,b1 + a,b2 + + anbn, and also the norms of these vectors are defined as Vā a = Važ + až + + a유, |||| = V5 .5 Vb+ b을 + + b. Prove the Cauchy-Schwarz inequality (a · b)? < |lä||2||b||°, that is the inequality (abı + azb2 + + anbn)² < (a² + a² + · + a) (bỉ + b3+. + b). Hint: Consider the function f(x) = (a1x + b1)² + (a2x + b2)² + . .. + (anx + br)² and apply (a).

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 (b)

(b) Let a = (a1, a2,..., an) and b = (b1, b2, ..., bn) be any two vectors in R". The inner product (dot product) of these two vectors are defined as a ·b = a,b1 + a,b2 + + anbn, and also the norms of these vectors are defined as Vā a = Važ + až + + a유, |||| = V5 .5 Vb+ b을 + + b. Prove the Cauchy-Schwarz inequality (a · b)? < |lä||2||b||°, that is the inequality (abı + azb2 + + anbn)² < (a² + a² + · + a) (bỉ + b3+. + b). Hint: Consider the function f(x) = (a1x + b1)² + (a2x + b2)² + . .. + (anx + br)² and apply (a).
Transcribed Image Text:(b) Let a = (a1, a2,..., an) and b = (b1, b2, ..., bn) be any two vectors in R". The inner product (dot product) of these two vectors are defined as a ·b = a,b1 + a,b2 + + anbn, and also the norms of these vectors are defined as Vā a = Važ + až + + a유, |||| = V5 .5 Vb+ b을 + + b. Prove the Cauchy-Schwarz inequality (a · b)? < |lä||2||b||°, that is the inequality (abı + azb2 + + anbn)² < (a² + a² + · + a) (bỉ + b3+. + b). Hint: Consider the function f(x) = (a1x + b1)² + (a2x + b2)² + . .. + (anx + br)² and apply (a).
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