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Lect3-03-web

# Lect3-03-web - MATH 304 Fall 2011 Linear Algebra Homework...

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MATH 304, Fall 2011 Linear Algebra

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Homework assignment #9 (due Thursday, November 10) All problems are from Leon’s 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.

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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

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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 ˆ 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 ˆ x is a least squares solution if and only if A T r x ) = 0 ⇐⇒ A T ( b A ˆ x ) = 0 ⇐⇒ A T A ˆ x = A T b . Corollary The normal system A T A x = A T b is always consistent.

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Problem. Find the constant function that is the least square fit to the following data x 0 1 2 3 f ( x ) 1 0 1 2 f ( x ) = c = c = 1 c = 0 c = 1 c = 2 = 1 1 1 1
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Lect3-03-web - MATH 304 Fall 2011 Linear Algebra Homework...

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