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Unformatted text preview: MATH 304, Fall 2011 Linear Algebra Homework assignment #9 (due Thursday, November 10) All problems are from Leons book (8th edition). Section 5.3: 1b, 1c, 3a, 5, 6 Section 5.4: 3b, 7c, 8a, 15b, 30 MATH 304 Linear Algebra Lecture 19: Least squares problems (continued). Norms and inner products. Orthogonal projection Theorem 1 Let V be a subspace of R n . Then any vector x R n is uniquely represented as x = p + o , where p V and o V . In the above expansion, p is called the orthogonal projection of the vector x onto the subspace V . Theorem 2 bardbl x v bardbl > bardbl x p bardbl for any v negationslash = p in V . Thus bardbl o bardbl = bardbl x p bardbl = min v V bardbl x v bardbl is the distance from the vector x to the subspace V . V V o p x Least squares solution System of linear equations: a 11 x 1 + a 12 x 2 + + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2 n x n = b 2 a m 1 x 1 + a m 2 x 2 + + a mn x n = b m A x = b For any x R n define a residual r ( x ) = b A x . The least squares solution x to the system is the one that minimizes bardbl r ( x ) bardbl (or, equivalently, bardbl r ( x ) bardbl 2 ). bardbl r ( x ) bardbl 2 = m summationdisplay i =1 ( a i 1 x 1 + a i 2 x 2 + + a in x n b i ) 2 Let A be an m n matrix and let b R m . Theorem A vector x is a least squares solution of the system A x = b if and only if it is a solution of the associated normal system A T A x = A T b . Proof: A x is an arbitrary vector in R ( A ), the column space of A . Hence the length of r ( x ) = b A x is minimal if A x is the orthogonal projection of b onto R ( A ). That is, if r ( x ) is orthogonal to R ( A ). We know that { row space } = { nullspace } for any matrix. In particular, R ( A ) = N ( A T ), the nullspace of the transpose matrix of A . Thus x is a least squares solution if and only if A T r ( x ) = A T ( b A x ) = A T A x = A T b ....
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This note was uploaded on 11/08/2011 for the course MATH 304 taught by Professor Hobbs during the Spring '08 term at Texas A&M.
 Spring '08
 HOBBS
 Linear Algebra, Algebra, Least Squares

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