(a) Use the general properties of an inner product (·, ·) on a vector space V, with induced norm ||u|| = V(u, u), to show the Cauchy-Schwarz inequality: For any u, v € V, |(u, v)| < || u|| ||v||- (CS) [Hint: For eacht e R, we have (u+tv, u+tv) > 0 (justify this). Expand this inner prod- uct to get an expression that is quadratic in t; show that the condition for this quadratic to remain no-negative for all t is equivalent to (CS). Of course for the geometric dot product, involving the cosine of an angle, (CS) can be shown very easily...] (b) For any norm || · || induced by an inner product, as in (a) above, prove the triangle inequality: For any u, v e V, ||u + v|| < ||u|| + ||v||, (A) with equality iff u and v are orthogonal. [Hint: Square both sides. Of course for || - || to be a norm it must satisfy (A); this result shows that for any inner product, the quantity /(u, u) indeed defines a norm.]

(a) Use the general properties of an inner product (·, ·) on a vector space V, with induced norm ||u|| = V(u, u), to show the Cauchy-Schwarz inequality: For any u, v € V, |(u, v)| < || u|| ||v||- (CS) [Hint: For eacht e R, we have (u+tv, u+tv) > 0 (justify this). Expand this inner prod- uct to get an expression that is quadratic in t; show that the condition for this quadratic to remain no-negative for all t is equivalent to (CS). Of course for the geometric dot product, involving the cosine of an angle, (CS) can be shown very easily...] (b) For any norm || · || induced by an inner product, as in (a) above, prove the triangle inequality: For any u, v e V, ||u + v|| < ||u|| + ||v||, (A) with equality iff u and v are orthogonal. [Hint: Square both sides. Of course for || - || to be a norm it must satisfy (A); this result shows that for any inner product, the quantity /(u, u) indeed defines a norm.]

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.1: Inner Product Spaces

Problem 44EQ

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