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

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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 .

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