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Unformatted text preview: Lecture11 1 October 11, 2007 Outline: 1) Quick Review of the Big Picture of the 4 subspaces 2) The picture for a Full Column rank matrix 3) Motivation for Projections (The small picture) 4) Projection onto a line 5) Projection onto a subspace and the "normal equations" A T Ax =A T b Lecture 11: Projections ^ Orthogonality of the 4 subspaces: The Big (Strangian) Picture Point: For every matrix A there are 4 fundamental subspaces Two in R n : Two in R m: The Big Picture of Ax =b Full Column Rank: Point: For every matrix A there are 4 fundamental subspaces Two in R n : Two in R m: The Small Picture of a Full Column rank problem: Motivation for Projections and "Least-Squares Problems" Lecture11 2 October 11, 2007 Projection onto a Line: (1-dim subspace) Projection onto a Line: (1-dim subspace) Steps: 1) find the "least squares solution" x 2) find the projection p 3) find the Projection Matrix P ^ Projection onto a Line: (1-dim subspace) Example: Project b =[ 1 1 1]' onto the line spanned by a...
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This note was uploaded on 06/02/2010 for the course APMA APMA E3101 taught by Professor Spiegelman during the Fall '07 term at Columbia.
- Fall '07