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Proposition 2.2.2 (Inner product properties). (IP1) The inner product is positive definite: (x,x) > 0 for all x E R", with equality if and only if x = 0.

Proposition 2.2.2 (Inner product properties). (IP1) The inner product is positive definite: (x,x) > 0 for all x E R", with equality if and only if x = 0.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 21E
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Question

Prove the inner product property (IP1)

Proposition 2.2.2 (Inner product properties). (IP1) The inner product is positive definite: (x, x) > 0 for all x E R", with equality if and only if x = 0. (IP2) The inner product is symmetric: (x, y) = (y, x) for all x, y E R". (IP3) The inner product is bilinear: (x + x', y) = (x, y) + (x', y), (ax, y) = a(x, y), (x, y + y') = (x, y) + (x, y'), (x, by) = b(x, y) for all a, b e R, x, x', y, y' E R".
Transcribed Image Text:Proposition 2.2.2 (Inner product properties). (IP1) The inner product is positive definite: (x, x) > 0 for all x E R", with equality if and only if x = 0. (IP2) The inner product is symmetric: (x, y) = (y, x) for all x, y E R". (IP3) The inner product is bilinear: (x + x', y) = (x, y) + (x', y), (ax, y) = a(x, y), (x, y + y') = (x, y) + (x, y'), (x, by) = b(x, y) for all a, b e R, x, x', y, y' E R".
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