AM025b_sample_test3_sol

AM025b_sample_test3_sol - THE UNIVERSITY OF WESTERN ONTARIO...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

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

Unformatted text preview: THE UNIVERSITY OF WESTERN ONTARIO DEPARTMENT OF APPLIED MATHEMATICS Applied Mathematics 025a Third Tutorial Test, November 22, 2007 Time: 60 minutes The test is closed book! You are allowed to use only simple calculators. Section 002, Professor Kiriushcheva 1 1. ( 10 marks ) Use the Gram-Schmidt process to convert the given basis S = 2- 1 1 , 3- 1 , 1 2 into an orthonormal basis. Sample solution: U 1 = (2, -1, 1) U 2 = (0, 3, -1) U 3 = (1, 2, 0) Using the Gram-Schmidt Process: (finding orthogonal basis) Step 1. V 1 = U 1 = (2, -1, 1) * || V 1 || = radicalBig (2) 2 + (- 1) 2 + (1) 2 = 6 || V 1 || 2 = 6 Step 2. V 2 = U 2- proj V 1 U 2 = U 2- < U 2 , V 1 > || V 1 || 2 V 1 = (0, 3, -1) - (- 4) 6 (2, -1, 1) V 2 = parenleftbigg 4 3 , 7 3 , - 1 3 parenrightbigg * || V 2 || = radicalBig ( 4 3 ) 2 + ( 7 3 ) 2 + (- 1 3 ) 2 = radicalBig 66 9 || V 2 || 2 = 66 9 Step 3. V 3 = U 3- proj V 1 U 3- proj V 2 U 3 = U 3- < U 3 , V 1 > || V 1 || 2 V 1- < U 3 , V 2 > || V 2 || 2 V 2 = (1, 2, 0) - 6 (2, -1, 1) - 6 66 / 9 parenleftbigg 4 3 , 7 3 , - 1 3 parenrightbigg V 3 = parenleftbigg- 1 11 , 1 11 , 3 11 parenrightbigg 2 * || V 3 || = radicalBig (- 1 11 ) 2 + ( 1 11 ) 2 + ( 3 11 ) 2 = radicalBig 1 11 Normalize (finding orthonormal basis) q 1 = V 1 || V 1 || = parenleftBigg 2 6 ,- 1 6 , 1 6 parenrightBigg q 2 = V 2 || V 2 || = parenleftBigg 4 66 , 7 66 ,- 1 66 parenrightBigg...
View Full Document

Page1 / 7

AM025b_sample_test3_sol - THE UNIVERSITY OF WESTERN ONTARIO...

This preview shows document pages 1 - 4. Sign up to view the full document.

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