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# handout6 - Math 171A Linear Programming Lecture 6 Full-Rank...

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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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handout6 - Math 171A Linear Programming Lecture 6 Full-Rank...

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