Lecture 7 - Properties of Incompatible Systems of Equations

Lecture 7 - Properties of Incompatible Systems of Equations...

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Lecture 7 Properties of incompatible systems UCSD Math 171A: Numerical Optimization Philip E. Gill ( pgill@ucsd.edu ) Wednesday, January 21, 2009 Question Given A R m × n , how do we characterize all vectors w R m such that w 6∈ range( A )? UCSD Center for Computational Mathematics Slide 2/34, Wednesday, January 21, 2009 y y = Ax x Set of all rhs vectors R n R m unreachable part of R m Notice that if w is not in range( A ), then w + y is not in range( A ) either (even if y range( A )!) UCSD Center for Computational Mathematics Slide 3/34, Wednesday, January 21, 2009 Example: A = 1 5 3 - 1 - 5 - 3 ! , rank( A ) = 1 range( A ) = ± y : y = α a 1 = α ² 1 - 1 ³ for some α ´ Observe that z = 1 1 ! 6∈ range( A ) because there is no α such that 1 1 ! = α 1 - 1 ! UCSD Center for Computational Mathematics Slide 4/34, Wednesday, January 21, 2009
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The vectors z = 1 1 ! and a 1 = 1 - 1 ! are independent. Moreover, they are orthogonal z T a 1 = ± 1 1 ² 1 - 1 ! = 0 But a 1 spans all of range( A ) z is orthogonal to every vector in range( A ) z satisfies A T z = 0 UCSD Center for Computational Mathematics Slide 5/34, Wednesday, January 21, 2009 Result Let z be a nonzero vector that is orthogonal to every column of A = ± a 1 a 2 ··· a n ² i.e., z T a j = 0 for j = 1, 2, . . . , n . Then z 6∈ range( A ). Proof: Let z range( A ) with z T a j = 0 for j = 1, 2, . . . , n . As z range( A ), there must be some y such that z = Ay , i.e., z = Ay = ± a 1 a 2 ··· a n ² y = n X j =1 a j y j If we form z T z , we get z T z = z T Ay = n X j =1 z T a j y j = n X j =1 ( z T a j ) y j = 0 UCSD Center for Computational Mathematics Slide 6/34, Wednesday, January 21, 2009 z T z = 0. but z T z = n j =1 z 2 j , so z = 0 z = 0 is the only vector in range( A ) with z T a j = 0. If we write the n identities z T a j = 0 as components of a vector: ( 0 0 ··· 0 ) = ( z T a 1 z T a 2 ··· z T a n ) = z T ( a 1 a 2 ··· a n ) = z T A Taking the transpose does not change this identity, so A T z = 0 UCSD Center for Computational Mathematics Slide 7/34, Wednesday, January 21, 2009 Definition (Null-space of a matrix) The set of z such that A T z = 0 is called the null space of A T . null(
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Lecture 7 - Properties of Incompatible Systems of Equations...

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