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AM025b_sample_test3_sol

# AM025b_sample_test3_sol - THE UNIVERSITY OF WESTERN ONTARIO...

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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...
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AM025b_sample_test3_sol - THE UNIVERSITY OF WESTERN ONTARIO...

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