sol55 - Partial Solution Set, Leon 5.5 5.5.2 (a) Straight...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Partial Solution Set, Leon § 5.5 5.5.2 (a) Straight forward computations (b) We have u 1 = ± 1 3 2 , 1 3 2 - 4 3 2 ² T , u 2 = ( 2 3 , 2 3 , 1 3 ) T , and u 3 = ± 1 2 , 1 2 , 0 ² T . Let x = (1 , 1 , 1) T . Write x as a linear combination of u 1 , u 2 , and u 3 , and use Parseval’s formula to compute || x || . Solution : We know from part (a) that [ u 1 , u 2 , u 3 ] is an orthonormal basis for R 3 . By Theorem 5.5.2, we know that x = ( x T u 1 ) u 1 + ( x T u 2 ) u 2 + ( x T u 3 ) u 3 = - 2 3 2 u 1 + 5 3 u 2 + 0 u 3 = - 2 3 2 u 1 + 5 3 u 2 By Parseval’s formula, || x || = ( 4 18 + 25 9 ) 1 / 2 = 3. 5.5.3 We are given S , the subspace spanned by u 2 and u 3 of the preceding exercise, and x = (1 , 2 , 2) T . We are to find the projection p of x onto S , and to verify that p - x S . Solution : The projection is p = ( x T u 2 ) u 2 + ( x T u 3 ) u 3 = 8 3 u 2 - 1 2 u 3 = ³ 23 18 , 41 18 , 8 9 ´ T So p - x = ( 5 18 , 5 18 , - 10 9 ) T . It is easy to show that p
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/17/2011 for the course MATH 311 taught by Professor Anshelvich during the Spring '08 term at Texas A&M.

Page1 / 3

sol55 - Partial Solution Set, Leon 5.5 5.5.2 (a) Straight...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online