A4_soln - Math 235 Assignment 4 Solutions 1 Consider P 2...

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Unformatted text preview: Math 235 Assignment 4 Solutions 1. Consider P 2 with inner product h p ( x ) ,q ( x ) i = p (- 1) q (- 1) + p (0) q (0) + p (1) q (1). (a) Find the value of h 1- x- x 2 , 1 + x 2 i . Solution: h 1- x- x 2 , 1 + x 2 i = (1)(2) + (1)(1) + (- 1)(2) = 1. (b) Find the distance between 1- x- x 2 and 1 + x 2 . Solution: The distance is || (1- x- x 2 )- (1 + x 2 ) || = || - x- 2 x 2 || = p (- 1) 2 + (0) 2 + (- 3) 2 = √ 10 . (c) Determine the coordinates of 1- 2 x + x 2 with respect to the orthonormal basis B = 1 √ 3 , 1 √ 2 x, 1 √ 6 (2- 3 x 2 ) Solution: We calculate the following inner products: 1 √ 3 , 1- 2 x + x 2 = 5 √ 3 1 √ 2 x, 1- 2 x + x 2 =- 4 √ 2 1 √ 6 (2- 3 x 2 ) , 1- 2 x + x 2 =- 2 √ 6 Therefore, [1- 2 x + x 2 ] B = 5 / √ 3- 4 / √ 2- 2 / √ 6 . (d) Given that S = { 1- x 2 , 1 2 ( x- x 2 ) } is orthonormal, extend S to find an orthonormal basis for P 2 ....
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A4_soln - Math 235 Assignment 4 Solutions 1 Consider P 2...

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