Sol-HW7 - Math Methods Solutions for Assignment 7 Oct 18...

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Unformatted text preview: Math Methods Solutions for Assignment 7 Oct. 18, 2010 Fall 2010 Linear Spaces, Linear Operators Due Oct. 25, 2010 1 (5). For the vectors in three dimensions, ~ v 1 = ˆ x + ˆ y, ~ v 2 = ˆ y + ˆ z, ~ v 3 = ˆ z + ˆ x use the Gram-Schmidt procedure to construct an orthonormal basis starting from ~ v 1 . ~e 1 = ~v 1 || v 1 || = ˆ x + ˆ y √ 2 ~u 2 = ~v 2- < ~e 1 | ~v 2 > ~e 1 ; < ~e 1 | ~v 2 > = 1 √ 2 ; ~u 2 = ˆ y + ˆ z- 1 √ 2 ˆ x + ˆ y √ 2 =- 1 2 ˆ x + 1 2 ˆ y + ˆ z ~e 2 = ~u 2 || u 2 || =- 1 2 ˆ x + 1 2 ˆ y + ˆ z q 1 4 + 1 4 + 1 =- 1 √ 6 ˆ x + 1 √ 6 ˆ y + r 2 3 ˆ z ~u 3 = ~v 3- < ~e 1 | ~v 3 > ~e 1- < ~e 2 | ~v 3 > ~e 2 ; < ~e 1 | ~v 3 > = 1 √ 2 ; < ~e 2 | ~v 3 > =- 1 √ 6 + r 2 3 = 1 √ 6 ~u 3 = ˆ z + ˆ x- ˆ x + ˆ y 2 + 1 6 ˆ x- 1 6 ˆ y- 1 3 ˆ z = 2 3 ˆ x- 2 3 ˆ y + 2 3 ˆ z = 2 3 (ˆ x- ˆ y + ˆ z ) ~e 3 = ~u 3 || u 3 || = 2 3 (ˆ x- ˆ y + ˆ z ) q 3 · 4 9 = ˆ x- ˆ y + ˆ z √ 3 2 (10). For the vector space of polynomials in x , use the scalar product defined as...
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Sol-HW7 - Math Methods Solutions for Assignment 7 Oct 18...

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