Q 5.5-21E Question: If ||v鈫抾| denotes th... [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: Q 5.5-21E (page 261)

Question: If $| | \overset{鈫�}{v} | |$ denotes the standard norm in Rⁿ does the formula
$鈱� \overset{鈫�}{v}, \overset{鈫�}{w} 鈱� = | | \overset{鈫�}{v} + \overset{鈫�}{w} | |^{2} - | | \overset{鈫�}{v} | | - | | \overset{鈫�}{w} | |$
define an inner product in Rⁿ ?

Short Answer

Expert verified

$鈱�\overset{鈫�}{v}, \overset{鈫�}{w}鈱� = 2 \overset{鈫�}{v} .2 \overset{鈫�}{w}$

Step by step solution

Inner product.

Inmathematics, an inner product space is areal vector spaceor acomplex vector spacewith anoperationcalled an inner product. The inner product of two vectors in the space is ascalar, often denoted withangle bracketssuch as in $鈱� a, b 鈱�$

For example,

Among the simplest examples of inner product spaces are R and C. The real numbers R are a vector space over R that becomes a real inner product space when endowed with standard multiplication as its real inner product:

$鈱�x, y鈱� = x \overset{鈫�}{y} f o r x, y 鈭� R$

Recall that,

$\begin{matrix} 鈱�\overset{鈫�}{v}, \overset{鈫�}{w}鈱� = | | \overset{鈫�}{v} + \overset{鈫�}{w} | |^{2} 鈭� | | \overset{鈫�}{v} | | 鈭� | | \overset{鈫�}{w} | |^{2} \\ = (\overset{鈫�}{v} + \overset{鈫�}{w}) . (\overset{鈫�}{v} + \overset{鈫�}{w}) 鈭� \overset{鈫�}{v} . \overset{鈫�}{v} 鈭� \overset{鈫�}{w} . \overset{鈫�}{w} \\ = \overset{鈫�}{v} . \overset{鈫�}{v} + \overset{鈫�}{v} . \overset{鈫�}{w} + \overset{鈫�}{w} . \overset{鈫�}{v} + \overset{鈫�}{w} . \overset{鈫�}{w} 鈭� \overset{鈫�}{v} . \overset{鈫�}{v} 鈭� \overset{鈫�}{w} . \overset{鈫�}{w} \\ = 2 \overset{鈫�}{v} . \overset{鈫�}{w} \end{matrix}$

Thus, $鈱�\overset{鈫�}{v}, \overset{鈫�}{w}鈱� = 2 \overset{鈫�}{v} .2 \overset{鈫�}{w}$

Hence, yes the formula define an inner product as $鈱�\overset{鈫�}{v}, \overset{鈫�}{w}鈱� = 2 \overset{鈫�}{v} .2 \overset{鈫�}{w}$

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91影视

Question: If $| | \overset{鈫�}{v} | |$ denotes the standard norm in Rⁿ does the formula
$鈱� \overset{鈫�}{v}, \overset{鈫�}{w} 鈱� = | | \overset{鈫�}{v} + \overset{鈫�}{w} | |^{2} - | | \overset{鈫�}{v} | | - | | \overset{鈫�}{w} | |$
define an inner product in Rⁿ ?

Short Answer

Step by step solution

Inner product.

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Question: If ||v鈫�|| denotes the standard norm in Rn does the formula鈱�v鈫�,w鈫�鈱�=||v鈫�+w鈫�||2-||v鈫�||-||w鈫�||define an inner product in Rn ?

Short Answer

Step by step solution

Inner product.

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Statistics

Logic and Functions

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Study anywhere. Anytime. Across all devices.