Question
please teach me the answer to #3.
The Cauchy-Schwarz Inequality Let u and v be unit vectors in R2. Our goal is to maximize u v. One way to do this is to interpret this as a constrained optimization problem: Note that |u| = 1 is equivalent to |u|2 = u u = 1. Let u = (x, y) and v = (z, w). Rewrite the above maximization problem in terms of x, y, z, w. Use Lagrange multipliers to show that u v is maximized provided u = v. Explain why the maximum value of u v must therefore be 1. Explain why for any unit vectors u and v we must have |u u| 1. Let u and v be any vectors in R2 (not necessarily unit). Apply your conclusion above to the vectors: to show that
Explanation / Answer
by 2 the maximum of u*v is when u=v
in this case, u*v=u*u=|u|^2=1, so the maximum is 1
