Chapter 9: Q. 43 (page 613)

Prove the polarization identity,
${||u + v||}^{2} - {||u - v||}^{2} = 4 (u \cdot v)$

Short Answer

Expert verified

${||u + v||}^{2} - {||u - v||}^{2} = 4 (u \cdot v)$

Step by step solution

Step 1. Given information

We have to prove the polarization identity

${||u + v||}^{2} - {||u - v||}^{2} = 4 (u \cdot v)$

Step 2. Proof

We can write ${||u + v||}^{2}$ as $(u + v) \cdot (u + v)$

and ${||u - v||}^{2}$ as $(u - v) \cdot (u - v)$

Now,

role="math" localid="1646800338985" ${||u + v||}^{2} - {||u - v||}^{2} = (u + v) \cdot (u + v) - (u - v) \cdot (u - v) = (u \cdot u + u \cdot v + v \cdot u + v \cdot v) - (u \cdot u - u \cdot v - v \cdot u + v \cdot v) = u \cdot u + u \cdot v + v \cdot u + v \cdot v - u \cdot u + u \cdot v + v \cdot u - v \cdot v = 2 (u \cdot v) + 2 (v \cdot u)$

Using commutative property, $(u \cdot v) = (v \cdot u)$

So,

$2 (u \cdot v) + 2 (v \cdot u) = 4 (u \cdot v)$

Therefore, ${||u + v||}^{2} - {||u - v||}^{2} = 4 (u \cdot v)$

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Prove the polarization identity,
${||u + v||}^{2} - {||u - v||}^{2} = 4 (u \cdot v)$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Proof

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Prove the polarization identity,u+v2-u-v2=4u·v

Short Answer

Step by step solution

Step 1. Given information

Step 2. Proof

One App. One Place for Learning.

Most popular questions from this chapter

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Logic and Functions

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Study anywhere. Anytime. Across all devices.

Prove the polarization identity,
${||u + v||}^{2} - {||u - v||}^{2} = 4 (u \cdot v)$