Logo
Home Study Material Contact Search
AI Chat FREE
  1. mathematics
  2. advance math
  3. the cauchy-schwarz inequality let u and v be vecto

The Cauchy-Schwarz Inequality Let u and v be vectors in R 2 . We wish to prove...

60.1K

Verified Solution

Question

Advance Math

The Cauchy-Schwarz Inequality Let u and v be vectors in R 2.

We wish to prove that ->    (u · v)^ 2 ? |u|^2 |v|^2 .

This inequality is called the Cauchy-Schwarz inequality and isone of the most important inequalities in linear algebra.

One way to do this to use the angle relation of the dot product(do it!). Another way is a bit longer, but can be considered anapplication of optimization. First, assume that the two vectors areunit in size and consider the constrained optimization problem:

Maximize u · v

Subject to |u| = 1 |v| = 1.

Note that |u| = 1 is equivalent to |u| 2 = u · u = 1.

(a) Let u = a b and v = c d . Rewrite the above maximizationproblem in terms of a, b, c, d.

(b) Use Lagrange multipliers to show that u · v is maximizedprovided u = v.

(c) Explain why the maximum value of u · v must, therefore, be1.

(d) Find the minimum value of u · v and explain why for any unitvectors u and v we must have |u · v| ? 1.

(e) Let u and v be any vectors in R 2 (not necessarily unit).Apply your conclusion above to the vectors: u |u| and v |v| to showthat (u · v) ^2 ? |u|^ 2 |v|^ 2 .

Answer & Explanation Solved by verified expert
3.8 Ratings (534 Votes)
    See Answer
Get Answers to Unlimited Questions

Join us to gain access to millions of questions and expert answers. Enjoy exclusive benefits tailored just for you!

Membership Benefits:
  • Unlimited Question Access with detailed Answers
  • Zin AI - 3 Million Words
  • 10 Dall-E 3 Images
  • 20 Plot Generations
  • Conversation with Dialogue Memory
  • No Ads, Ever!
  • Access to Our Best AI Platform: Zin AI - Your personal assistant for all your inquiries!
Become a Member

Other questions asked by students

Q

A certain commercial mass spectrometer is used to separate uranium ions of mass 3.92 × 10-25...

Physics
Q

One unique aspect of I/O psychology is its double-sided nature of being both a science and...

Psychology
Q

Which part of the sampleintroduction is Section 2?A. This is the hook that gets the...

Biology
Q

Exercise 5C: Utility of a restricted resource The School Pack company manufactures bags for school...

Accounting
Q

TB MC Qu. 17-97 (Algo) Kay Company budgets overhead... requiring 5 direct labor hours? Multiple...

Accounting
Q

Using the general principles of ordinary income, is income proceeds from selling the copyright to...

Accounting
Q

Water Utility Fund 14. The Dwyer Water Utility Fund billed nongovernmental customers for $300,000 and...

Accounting
Q

The cost of goods sold computations for Crane Company and Swifty Company are shown below....

Accounting
Q

The following is a manufacturing cost report of Marching Ants Inc.Marching Ants Inc.Manufacturing CostsFor the...

Accounting
Q

Exercise 1. (10 points) Selected financial information from Syvitex Corporation is reproduced below. NOA tumorer...

Finance
Q

a 20 years $1,000 par value bond has a 7% annual payment coupon. the bond...

Finance
Q

What is the total risk of the returns on an asset that gives returns of...

Finance