Explanation Approach: If u = ( u 1 , u 2 , ... , u n ) and v = ( v 1 , v 2 , ... , v n ) are vectors in ℂ n , then the standard inner product in ℂ n is given by, 〈 u , v 〉 = u 1 v 1 ¯ + u 2 v 2 ¯ + ... + u n v n ¯ . The norm of v is defined as, | | v | | = 〈 v , v 〉 That is, | | v | | 2 = 〈 v , v 〉 The all the properties which need to be verified on inner product on ℂ n are given as, 1. 〈 v , v 〉 ≥ 0 , furthermore 〈 v . v 〉 = 0 if and only if v = 0 . 2. 〈 w , v 〉 = 〈 v , w 〉 ¯ 3. 〈 k v , w 〉 = k 〈 v , w 〉 4. 〈 u , v + w 〉 = 〈 u , v 〉 + 〈 u , w 〉 Here, vector v , w and u in ℂ n and k is any scalar number. Calculation: Consider the expression, | | v + w | | 2 As, | | v | | 2 = 〈 v , v 〉 , the above is expressed as, | | v + w | | 2 = 〈 v + w , v + w 〉 Now, as 〈 u , v + w 〉 = 〈 u , v 〉 + 〈 u , w 〉 , the above expression is expressed as, | | v + w | | 2 = 〈 v , v + w 〉 + 〈 w , v + w 〉 = 〈 v + w , v 〉 ¯ + 〈 v + w , w 〉 ¯ = 〈 v , v 〉 + 〈 w , v 〉 ¯ + 〈 v ,

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

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Chapter 5.1, Problem 37P

To determine

To prove:

The complex inner product space V, for all v,w∈V, ||v+w||2=||v||2+2Re(〈v,w〉)+||v||2, where Re denotes the real part of a complex number.

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Chapter 5.1, Problem 36P

Chapter 5.2, Problem 1P

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Chapter 5 Solutions

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To prove: The complex inner product space V , for all v , w ∈ V , | | v + w | | 2 = | | v | | 2 + 2 Re ( 〈 v , w 〉 ) + | | v | | 2 , where Re denotes the real part of a complex number .

Solution Summary: The author explains how to prove the complex inner product space V.

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To prove:

The complex inner product space V, for all v,w∈V, ||v+w||2=||v||2+2Re(〈v,w〉)+||v||2, where Re denotes the real part of a complex number.

Want to see the full answer?

Chapter 5 Solutions

To prove: The complex inner product space V , for all v , w ∈ V , | | v + w | | 2 = | | v | | 2 + 2 Re ( 〈 v , w 〉 ) + | | v | | 2 , where Re denotes the real part of a complex number .

Solution Summary: The author explains how to prove the complex inner product space V.

Videos

To prove: The complex inner product space V, for all v,w∈V, ||v+w||2=||v||2+2Re(〈v,w〉)+||v||2, where Re denotes the real part of a complex number.

Want to see the full answer?

Chapter 5 Solutions

To prove:

The complex inner product space V, for all v,w∈V, ||v+w||2=||v||2+2Re(〈v,w〉)+||v||2, where Re denotes the real part of a complex number.