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Unformatted text preview: Lecture 09 1 October 01, 2007 Outline: 1) The general solution to Ax =b 2) The BIG ideas A) Linear Independence B) Span C) Basis D) Dimension 3) Bases and Dimensions for the 4 Fundamental Subspaces of a mxn Matrix A C(A), N(A), C(A T ), N(A T ) 4) Orthogonality of the 4 subspaces Lecture 09: The Big picture The General solution to Ax =b Given A in R mxn and b in R m , we want to find all solutions to Ax=b (if they exist) General Algorithm: 1) use Gauss-Jordan Elimination to reduce [ A b ] to [R d ] 2) determine if the rhs is consistent (i.e. d in C(R) or b in C(A)) A) if not consistent, there is no solution B) if consistent then find one particular solution such that Rx p =d (or Ax p =b ) (combination of pivot columns of R that add to form d) 3) Find all special solutions to Ax =0 and form null space matrix N 4) the general solution is x =x p +x N where x N =Nc is all vectors in the Null space N(A) The General solution to Ax =b Example #1: A=[ 1 2 1 0 1 ; 2 4 1 0 0; 1 2 0 1 -4], b =[ 1 1 1]' The General solution to Ax =b Example #2: A=[ 1 1 ; 1 -2 ; -2 1 ], b =[ b 1 b 2 b 3 ]' Lecture 09 2 October 01, 2007 The General solution to Ax =b Types of solutions......
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- Fall '07