2009bsol3 - M346 Third Midterm Exam Solutions 1 Gram Schmidt a(10 points On R 3 with the usual inner product Use Gram-Schmidt to convert x 1 =(1 2

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Unformatted text preview: M346 Third Midterm Exam Solutions, November 20, 2009 1) Gram Schmidt: a)(10 points) On R 3 with the usual inner product, Use Gram-Schmidt to convert x 1 = (1 , 2 , 0) T , x 2 = (3 , 1 , 1) T , x 3 = (4 , 3 , − 5) T to an orthogonal basis. y 1 = x 1 = 1 2 y 2 = x 2 − ( y 1 | x 2 ) ( y 1 | y 1 ) y 1 = 3 1 1 − 5 5 1 2 = 2 − 1 1 y 3 = x 3 − ( y 1 | x 3 ) ( y 1 | y 1 ) y 1 − ( y 2 | x 3 ) ( y 2 | y 2 ) y 2 = 4 3 − 5 − 10 5 1 2 − 6 2 − 1 1 = 2 − 1 − 5 b)(15 points) On R 2 [ t ] with the inner product ( f | g ) = integraltext 2 f ( t ) g ( t ) dt , trans- form { 1 , t, t 2 } to an orthogonal basis. y 1 = x 1 = 1 y 2 = x 2 − integraltext 2 t dt integraltext 2 1 dt y 1 = t − 1 y 3 = x 3 − integraltext 2 t 2 dt integraltext 2 1 dt y 1 − integraltext 2 t 2 ( t − 1) dt integraltext 2 ( t − 1) 2 dt y 2 = t 2 − 4 3 − 2( t − 1) = t 2...
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This note was uploaded on 04/10/2011 for the course M 346 taught by Professor Radin during the Spring '08 term at University of Texas at Austin.

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2009bsol3 - M346 Third Midterm Exam Solutions 1 Gram Schmidt a(10 points On R 3 with the usual inner product Use Gram-Schmidt to convert x 1 =(1 2

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