hw13 - { u,v } of Span( { x,y } ) such that x is a scalar...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 110—Linear Algebra Fall 2009, Haiman Problem Set 13 Due Friday, Dec. 4, at the beginning of lecture. 1. In the textbook and lecture we saw a quick but “tricky” proof of the Cauchy-Schwarz inequality, Theorem 6.2(c). In this problem we will work out a slightly longer but easier to motivate proof using Gram-Schmidt. Note that the proof of Theorem 6.4, which justifies the Gram-Schmidt process, does not use the Cauchy-Schwarz or triangle inequalities, so it is not circular reasoning to use Gram-Schmidt to prove Cauchy-Schwarz. (a) If x and y are linearly dependent, then they are both scalar multiples of some vector u . Verify that Cauchy-Schwarz holds with equality in this case. (b) If x and y are linearly independent, deduce from the Gram-Schmidt process that there is an orthonormal basis
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: { u,v } of Span( { x,y } ) such that x is a scalar multiple of u . Letting x = tu , y = au + bv , express h x,y i , k x k , and k y k in terms of t , a , and b , and use this to verify that the Cauchy-Schwarz inequality holds, with strict inequality in this case. 2. Prove that if f : [ a,b ] C is a continuous function, then Z b a f ( t ) dt s ( b-a ) Z b a | f ( t ) | 2 dt Hint: nd a way to apply Cauchy-Schwarz. 3. Section 6.1, Exercises 28 and 29. 4. Section 6.2, Exercise 2(g). 5. Section 6.2, Exercise 21. 6. Use Theorem 6.6 to prove that if W is a nite-dimensional subspace of an inner product space V , then ( W ) = W . 7. Section 6.3, Exercise 20(b)....
View Full Document

Ask a homework question - tutors are online