A4 - k k 2 = k ~v 1 k 2 + + k ~v k k 2 4. Let V be a nite...

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Math 235 Assignment 4 Due: Wednesday, June 1st 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 . (b) Find the distance between 1 - x - x 2 and 1 + x 2 . (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 ) ² (d) Given that S = { 1 - x 2 , 1 2 ( x - x 2 ) } is orthonormal, extend S to find an orthonormal basis for P 2 . 2. Let V be an inner product space with inner product h , i . Prove that for any ~v V and t R we have: (a) h ~v, ~ 0 i = 0 (b) k t~v k = | t |k ~v k 3. Prove that if { ~v 1 ,...,~v k } is an orthogonal set in an inner product space V , then k ~v 1 + ··· + ~v
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Unformatted text preview: k k 2 = k ~v 1 k 2 + + k ~v k k 2 4. Let V be a nite dimensional inner product space. Let L : V V be a linear operator. (a) If B = { ~v 1 ,...,~v n } is an orthonormal basis for V , then prove that the ij-th entry of [ L ] B is h L ( ~v j ) ,~v i i . (b) Prove that if h ~v,L ( ~w ) i = h L ( ~v ) , ~w i for all ~v, ~w V , then for any orthonormal basis C of V , we have that [ L ] C is symmetric (that is, ([ L ] C ) ij = ([ L ] C ) ji )....
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