Lecture 6 - Full-Rank Systems of Equations

Lecture 6 - Full-Rank Systems of Equations - Lecture 6...

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Unformatted text preview: Lecture 6 Full-Rank Systems of Equations UCSD Math 171A: Numerical Optimization Philip E. Gill ( [email protected] ) Friday, January 16th, 2009 Definition If r = rank( A ), then any subset of r linearly independent columns of A is known as a column basis (or just basis ) for range( A ). A column basis is defined by r column indices { β 1 , β 2 , . . ., β r } with B = ( a β 1 a β 2 ··· a β r | {z } column basis ) UCSD Center for Computational Mathematics Slide 2/32, Friday, January 16th, 2009 A B m × r UCSD Center for Computational Mathematics Slide 3/32, Friday, January 16th, 2009 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 4/32, Friday, January 16th, 2009 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 5/32, Friday, January 16th, 2009 Adding a dependent column to a matrix does not change the 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 6/32, Friday, January 16th, 2009 Now assume that b 6∈ range( A ) ⇒ b is not a linear combination of the columns of A ⇒ b is independent of the columns of A ⇒ Adding an independent column increases the rank ⇒ rank ( A b ) = rank( A ) + 1 > rank( A ) UCSD Center for Computational Mathematics Slide 7/32, Friday, January 16th, 2009 Example 1: A = 1 1 1 1- 1 2 1 !...
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This note was uploaded on 10/23/2010 for the course MATH 171a taught by Professor Staff during the Winter '08 term at UCSD.

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Lecture 6 - Full-Rank Systems of Equations - Lecture 6...

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