lina-13-11-09-s1

# lina-13-11-09-s1 - Linear Algebra Geometry Sheet 6 Set on...

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Linear Algebra Geometry: Sheet 6 Set on Friday, November 12: Questions 1, 2, and 3 1. Let V R n be a linear subspace and u 1 , u 2 , ··· , u k an orthonormal basis of V . (a) For x R n we say that x V if for any v V we have x · v = 0. Show that x V is equivalent to x · u i for i = 1 , 2 , ··· ,k . (b) Show that P V x = 0 is equivalent to x V . (c) Set V := { x R n ; x V } . Show that V is a linear subspace of R n . (d) Show that for any x R n P V x + P V x = x . (Hint: Show that an orthonormal basis of V and an orthonormal basis of V together form an orthonormal basis of R n .) (e) Show that the distance of x R n to V is given by k P V x k . (f) Let V R 10 be the hyperplane perpendicular to the vector w = (0 , - 1 , 0 , 3 , 0 , 5 , 0 , - 1 , 0 , 8). Compute the distance of x = (0 , 0 , 1 , 1 , - 8 , 12 , 1 , 0 , 0 , - 3) to the hyperplane V . 2. Gram Schmidt process: Let w 1 , w 2 , w 3 R 3 be linearly independent. Set v 1 := w 1 k w 1 k V 1 := span { v 1 } v 2 := w 2 - P V 1 w 2 k w 2 - P V 1 w 2 k V 2
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## This note was uploaded on 02/25/2010 for the course MATHEMATIC 11007 taught by Professor Spyros during the Spring '10 term at Bristol Community College.

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