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

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 is one 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 an application of optimization. First, assume that the two vectors are unit 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 maximization problem in terms of a, b, c, d.

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

(c) Explain why the maximum value of u · v must, therefore, be 1.

(d) Find the minimum value of u · v and explain why for any unit vectors 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 show that (u · v) ^2 ≤ |u|^ 2 |v|^ 2 .

Expert Solution

Colby Messinger answered 1 year ago

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