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**Unformatted text preview: **Math 171A: Linear Programming Lecture 6 Full-Rank Systems of Linear Equations Philip E. Gill c 2011 http://ccom.ucsd.edu/~peg/math171a Friday, January 14th, 2011 Recap: Two fundamental subspaces range( A ) 4 = { y : y = Ax for some x R n } range( A T ) 4 = { x : x = A T y for some y R m } The set range( A ) lives in R m , i.e., range( A ) R m The set range( A T ) lives in R n , i.e., range( A T ) R n UCSD Center for Computational Mathematics Slide 2/35, Friday, January 14th, 2011 y y = Ax x R n R m range( A ) y x R n R m range( A T ) x = A T y Recap: Two fundamental subspaces range( A ) 4 = { y : y = Ax for some x R n } range( A T ) 4 = { x : x = A T y for some y R m } column rank( A ) = maximum number of independent columns of A row rank( A ) = maximum number of independent rows of A column rank( A ) n and row rank( A ) m UCSD Center for Computational Mathematics Slide 4/35, Friday, January 14th, 2011 Result row rank( A ) = column rank( A ) The common value is the rank of A , denoted by rank( A ). row rank( A ) m column rank( A ) n rank( A ) min( m , n ) UCSD Center for Computational Mathematics Slide 5/35, Friday, January 14th, 2011 Question Given A = ( a 1 a 2 a n ) R m n and b R m , how can we tell if b range( A )? UCSD Center for Computational Mathematics Slide 6/35, Friday, January 14th, 2011 First, assume that the system Ax = b is compatible b range( A ) b = Ax for some x n X j =1 a j x j- b = 0 the vectors { a 1 , a 2 , . . ., a n , b } must be dependent A b has dependent columns UCSD Center for Computational Mathematics Slide 7/35, Friday, January 14th, 2011 Adding a dependent column to a matrix does not change its rank. The maximal number of linearly independent columns of A b = a 1 a 2 a n b is the same as the maximal number of linearly independent columns of A = ( a 1 a 2 a n ) rank A b = rank( A ) UCSD Center for Computational Mathematics Slide 8/35, Friday, January 14th, 2011 Now assume that b 6 range( A ) b is not a linear combination of the columns of...

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