Ch. 6: Lec. 25 The fundamental theorem of linear algebra Approximating matrices with SVD The basic idea Guess who? Bonus example 1 Bonus example 2 Frame 1/16 Chapter 6: Lecture 25 Linear Algebra, Course 124B, Fall, 2008 Prof. Peter Dodds Department of Mathematics & Statistics University of Vermont Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. Ch. 6: Lec. 25 The fundamental theorem of linear algebra Approximating matrices with SVD The basic idea Guess who? Bonus example 1 Bonus example 2 Frame 2/16 Outline The fundamental theorem of linear algebra Approximating matrices with SVD The basic idea Guess who? Bonus example 1 Bonus example 2 Ch. 6: Lec. 25 The fundamental theorem of linear algebra Approximating matrices with SVD The basic idea Guess who? Bonus example 1 Bonus example 2 Frame 3/16 All the way with A x = b: Applies to any m × n matrix A. Symmetry of A and A T . Where x lives: Row space C(A T ) ⊂ R n . (Right) Nullspace N (A) ⊂ R n . dim C(A T ) + dim N (A) = r +(n - r )= n Orthogonality: C(A T ) N (A)= R n Where b lives: Column space C(A) ⊂ R m . Left Nullspace N (A T ) ⊂ R m . dim C(A) + dim N (A T ) = r +(m - r )= m Orthogonality: C(A) N (A T )= R m Ch. 6: Lec. 25 The fundamental theorem of linear algebra Approximating matrices with SVD The basic idea Guess who? Bonus example 1 Bonus example 2 Frame 4/16 Best solution x * when b = p + e: Space Left Null R m 0 0 d = m - r d = r Row Space Column Space R n Null Space x n x r A x n = 0 A x r = p Ax * = p d = n - r d = r x * = x r + x n p b e
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Outline Chapter 6: Lecture 25 - University of Vermontpdodds/teaching/courses/2008-08UVM-124/docs/... · 2009-01-14 · Chapter 6: Lecture 25 Linear Algebra, Course 124B, Fall, 2008
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