(a) Let a,..., a, and by,...,b be real numbers. Prove the Schwarz inequality. That is,
(b) When does equality hold in the Schwarz inequality?
(a) Let a,..., a, and by,...,b be real numbers. Prove the Schwarz inequality. That is,
#2. Let n E N and X1,X2, ,yn, and zi,22, An be real numbers. ,An, yī,Y2, #a) Prove the identity #b) Use the identity in #a) to prove (the Cauchy-Schwartz inequality) that #1) Extend the result in #b) to prove that #d) Use the inequality in #b) to prove the inequality which is the triangle inequality
#2. Let n E N and X1,X2, ,yn, and zi,22, An be real numbers. ,An, yī,Y2, #a) Prove the identity #b) Use the identity...
#2. Let n E N and x1,x2,.., Xn, yı,y2,..,Ja, and zł,Zy, #a) Prove the identity An be real numbers #b) Use the identity in #a) to prove (the Cauchy-Schwartz inequality) that #1) Extend the result in #b) to prove that 4 #d) Use the inequality in #b) to prove the inequality which is the triangle inequality
#2. Let n E N and x1,x2,.., Xn, yı,y2,..,Ja, and zł,Zy, #a) Prove the identity An be real numbers #b) Use the identity in...
For this chapter the Cauchy-Schwarz inequality can be used (without a proof). (C – S) |u7v|2 < (uTu)(v7v) = ||_||- ||0||?, for all u, v ER", with equality only if u and v are parallel (u= lv, for some scalar 1). For the next three problems, let B be a symmetric and positive definite nxn matrix, and let geRn be a nonzero column vector. Define pu = - gʻg -9, and pB = -B-19. g? Bg P1) Prove that ||...
VII (5) (a) Prove the Cauchy-Schwarz inequality for vectors in R”: v•w |v||w| for all v, w ER. Also show that equality holds if and only if v = lw for some > 0. HINT: Assume, without loss of generality, that v, w # 0. Consider the non-negative function o(t) = \v – tw|2. Show that º attains a minimum at t = 6:12. Evaluate o at this point and use the fact that ¢ is non-negative to conclude. Address...
Prove that for positive real numbers x and y, the following inequality holds:
Show that if x and y are real numbers, x2 + y2 >= 2xy and
(x + y)2 >= 4xy; When does equality hold (with proof)?
Show that if x and y are real numbers, x2 + y2 2xy and (x y) 2Hry. When does equality hold (with proof)?
5. Let X be a non-negative integer-valued random variable with positive expectation. Prove that (Hint: Use the following special case of the Cauchy-Schwarz Inequality: First, make sure you see why this is a special case of the Cauchy-Schwarz Inequality; then apply it to get one of the inequalities of this problem.)
5. Let X be a non-negative integer-valued random variable with positive expectation. Prove that (Hint: Use the following special case of the Cauchy-Schwarz Inequality: First, make sure you see...
6) One of the most powerful inequalities in mathematics is the Cauchy-Bunyakovsky-Schwarz (CBS) inequality. It states: if u, u E R", then lü.히< 111||1| Further, equality is achieved only when ü and v are parallel, i.e. when u = cu for some constant c. Here are several problems that explore the power and scope of CBS. a) Suppose you want to maximize f(x, y, 2) = 8x + 4y +z subject to +y2 + z2 = 1. One way is...
5. Let X be a non-negative integer-valued random variable with positive expectation. Prove that E X2] (Hint: Use the following special case of the Cauchy-Schwarz Inequality: First, make sure you see why this is a special case of the Cauchy-Schwarz Inequality; then apply it to get one of the inequalities of this problem.)
5. Let X be a non-negative integer-valued random variable with positive expectation. Prove that E X2] (Hint: Use the following special case of the Cauchy-Schwarz Inequality: First,...
5. Let X be a non-negative integer-valued random variable with positive expectation. Prove that E X2] (Hint: Use the following special case of the Cauchy-Schwarz Inequality: First, make sure you see why this is a special case of the Cauchy-Schwarz Inequality; then apply it to get one of the inequalities of this problem.)
5. Let X be a non-negative integer-valued random variable with positive expectation. Prove that E X2] (Hint: Use the following special case of the Cauchy-Schwarz Inequality: First,...