08bAss4 - AS4/MATH1111/YKL/08-09 THE UNIVERSITY OF HONG...

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AS4/MATH1111/YKL/08-09 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1111: Linear Algebra Assignment 4 Due date : Apr 16, 2008 before 6:30 p.m. Where to hand-in : Assignment Box outside the lifts on the 4th floor of Run Run Shaw Remember to write down your Name , Uni. no. and Tutorial Group number . If you find difficulties, you are welcome to see the instructor, tutors or seek help from the help room. See “Information” at http://147.8.101.93/MATH1111/ for availabilities. Normally we do not count assignment grades in your final score. Nevertheless, any discovered plagiarism will be referred to University Disciplinary Committee. 1. Let A R m × n and B R n × r be two matrices. (a) Show that the nullity of AB is greater than or equal to the nullity of B . (b) Show that if rank( A ) = n , then rank( B ) = rank( AB ). 2. (a) Let A = 1 1 2 0 1 1 1 3 4 . Show that R 3 = N ( A ) c ( A ), where N ( A ) is the nullspace and c ( A ) is the column space of A . Is it always true that R 3 = N ( B ) + c ( B ) for any 3 × 3 matrix B ?
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