Homework 3 Solutions

Homework 3 Solutions - Math 171A: Mathematical Programming...

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Unformatted text preview: Math 171A: Mathematical Programming Instructor: Philip E. Gill Winter Quarter 2009 Homework Assignment #3 Due Monday February 2, 2009 Note the extended due date. Starred exercises require the use of Matlab . Exercise 3.1. Let A denote an m n matrix. (a) If A has rank m , what does this imply about the relative sizes of m and n ? A full-rank matrix has rank( A ) = min ( m,n ) . If A has rank m , then it must hold that m n . (b) Answer part (a) , this time assuming that A has rank n . If A satisfies rank( A ) = min ( m,n ) = n then n m . (c) Show that when A has rank n , any solution (if it exists) of Ax = b is unique. We have that n m from part (b) . If x 1 and x 2 are two solutions to the system Ax = b , then Ax 1 = b and Ax 2 = b . Subtracting these last two equations gives Ax 1- Ax 2 = 0 , or A ( x 1- x 2 ) = 0 . Hence x 1- x 2 null( A ) . The rank of A is n , which implies that n columns of A are linearly independent. If the columns of A are linearly independent then a linear combination Av of these columns can be zero only if v is zero (which is the same as saying that A must have the trivial null-space consisting of only the zero vector). As A ( x 1- x 2 ) = 0 is a zero linear combination, it follows that x 1- x 2 = 0 , which implies that x 1 = x 2 . Exercise 3.2....
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Homework 3 Solutions - Math 171A: Mathematical Programming...

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