Homework 3 Solutions

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

Page1 / 5

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online