lecture05 - Math 19b Linear Algebra with Probability Oliver...

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Math 19b: Linear Algebra with ProbabilityOliver Knill, Spring 2011Lecture 5: Gauss-Jordan eliminationWe have seen in the last lecture that a system of linear equations likevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglex+y+z= 3x-y-z= 5x+ 2y-5z= 9vextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglecan be written in matrix form asAvectorx=vectorb, whereAis amatrixcalledcoefficient matrixandcolumn vectorsvectorxandvectorb.A=1111-1-112-5,vectorx=xyz,vectorb=359.Thei’th entry (Avectorx)iis the dot product of thei’th row ofAwithvectorx.Theaugmented matrixis matrix, where other column has been added. This column containsthe vectorb. The last column is often separated with horizontal lines for clarity reasons.B=111|31-1-1|512-5|9.We will solve this equation using Gauss-Jordanelimination steps.1We aim is to find all the solutions to the system of linear equationsvextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglex+u=3y+v=5z+w=9x+y+z=8u+v+w=9vextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsingle.zyxwvuThis system appears intomographylike magnetic resonance

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