Tag Archives: Schwarz Inequality

Linear Algebra and Its Applications, Exercise 3.2.7

Posted on May 11, 2014 by hecker

Exercise 3.2.7. Show that . Hint: use the Schwarz inequality with an appropriate choice of . Answer: As noted in the hint, the key to proving this is to find an appropriate choice of . The easiest way to do … Continue reading →

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Linear Algebra and Its Applications, Exercise 3.2.6

Posted on May 10, 2014 by hecker

Exercise 3.2.6. Suppose that and are unit vectors. Then a one-line proof of the Schwarz inequality is as follows: What previous exercise justifies the middle step of this proof? Answer: From exercise 3.2.1(a) we have for any positive and .  … Continue reading →

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Linear Algebra and Its Applications, Exercise 3.2.4

Posted on May 3, 2014 by hecker

Exercise 3.2.4. Show that the Schwarz inequality is an equality if and only if and are on the same line through the origin. Describe the situation if and are on the opposite sides of the origin. Answer: We assume that … Continue reading →

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Linear Algebra and Its Applications, Exercise 3.2.1

Posted on April 20, 2014 by hecker

Exercise 3.2.1. a) Consider the vectors and where and are arbitrary positive real numbers. Use the Schwarz inequality involving and to derive a relationship between the arithmetic mean and the geometric mean . b) Consider a vector from the origin … Continue reading →

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