Q12E Give an algebraic proof for the ... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q12E (page 215)

Give an algebraic proof for the triangle inequality
$| | \overset{鈬赌}{v} + \overset{鈬赌}{w} | | 鈮� | | \overset{鈬赌}{v} | | + | | \overset{鈬赌}{w} | |$
Draw a sketch. Hint: Expand ${| |\overset{鈬赌}{v} + \overset{鈬赌}{w}| |}^{2} = (\overset{鈬赌}{v} + \overset{鈬赌}{w}) 路 (\overset{鈬赌}{v} + \overset{鈬赌}{w})$
Then use the Cauchy鈥揝chwarz inequality.

Short Answer

Expert verified

The triangle inequality is confirmed.

Step by step solution

Define Cauchy–Schwarz inequality.

If $\overset{鈬赌}{x}$ and $\overset{鈬赌}{y}$ are vectors in Rⁿthen

$| \overset{鈬赌}{x} 路 \overset{鈬赌}{y} | 鈮� | |\overset{鈬赌}{x}| | 路 | |\overset{鈬赌}{y}| |$

This statement is an equality if (and only if) $\overset{鈬赌}{x}$ and $\overset{鈬赌}{y}$ are parallel.

Determine the triangle inequality and draw the sketch

Use the hint:

${| | \overset{鈬赌}{v} + \overset{鈬赌}{w} | |}^{2} = | | \overset{鈬赌}{v} | |^{2} + {| | \overset{鈬赌}{w} | |}^{2} + 2 (\overset{鈬赌}{v} 路 \overset{鈬赌}{w}) {| | \overset{鈬赌}{v} + \overset{鈬赌}{w} | |}^{2} 鈮� | | \overset{鈬赌}{v} | |^{2} + {| | \overset{鈬赌}{w} | |}^{2} + 2 | | \overset{鈬赌}{v} | | | | \overset{鈬赌}{w} | | {| | \overset{鈬赌}{v} + \overset{鈬赌}{w} | |}^{2} = {(| | \overset{鈬赌}{v} | | + | | \overset{鈬赌}{w} | |)}^{2} {| | \overset{鈬赌}{v} + \overset{鈬赌}{w} | |}^{2} 鈮� {(| | \overset{鈬赌}{v} | | + | | \overset{鈬赌}{w} | |)}^{2}$

The sketch for the inequality is shown below:

Thus, by taking square root of both sides, triangle inequality is confirmed.

Hence, the triangle inequality is confirmed.

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91影视

Short Answer

Step by step solution

Define Cauchy–Schwarz inequality.

Determine the triangle inequality and draw the sketch

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91影视

Give an algebraic proof for the triangle inequality||v鈬赌+w鈬赌||鈮�||v鈬赌||+||w鈬赌||Draw a sketch. Hint: Expand|v鈬赌+w鈬赌|2=(v鈬赌+w鈬赌)路(v鈬赌+w鈬赌)Then use the Cauchy鈥揝chwarz inequality.

Short Answer

Step by step solution

Define Cauchy–Schwarz inequality.

Determine the triangle inequality and draw the sketch

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Discrete Mathematics

Calculus

Applied Mathematics

Probability and Statistics

Statistics

Study anywhere. Anytime. Across all devices.