math102HW4

math102HW4 - Math 102 - Homework 4 (selected problems)...

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Unformatted text preview: Math 102 - Homework 4 (selected problems) David Lipshutz Problem 1. (Strang, 3.2: #14) What matrix P projects every point in R 3 onto the line on intersection of the planes x + y + t = 0 and x- t = 0? Proof. The line of intersection must satisfy, t = x and y =- x- t =- 2 x , so the line can be written a = x y t = x- 2 x x = 1- 2 1 x Alternatively, the projection is onto the nullspace of A : A = " 1 1 1 1 0- 1 # , a = Null( A ) = 1- 2 1 The projection matrix is given by: P = aa T a T a = 1 6 1- 2 1 [ 1- 2 1 ] = 1 / 6- 1 / 3 1 / 6- 1 / 3 2 / 3- 1 / 3 1 / 6- 1 / 3 1 / 6 Problem 2. (Strang, 3.3: #6) Find the projection of b onto the column space of A : A = 1 1 1- 1- 2 4 , b = 1 2 7 Proof. The projection of b onto the column space of A is given by p = A ( A T A )- 1 A T b = 1 1 1- 1...
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math102HW4 - Math 102 - Homework 4 (selected problems)...

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