A1_soln

# A1_soln - Math 235 Assignment 1 Solutions 1. Let A be an m...

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Math 235 Assignment 1 Solutions 1. Let A be an m × n matrix and B be an n × p matrix. a) Prove that rank( AB ) rank( A ). Solution: Since the rank of a matrix is equal to the dimension of its column space, we consider the column space of A and AB . Observe that if ~ b Col( AB ), then there exists ~x R p such that AB~x = ~ b . Hence, if we let ~ y = B~x , we get A~ y = ~ b and so ~ b is also in the column space of A . Thus, the column space of AB is a subspace of the column space of A . Hence, the dimension of the column space of AB must be less than or equal to the dimension of the column space of A which is the desired result. b) Prove that rank( AB ) rank( B ). Solution: The null space of B is a subspace of the null space of AB , because if B~x = ~ 0, then AB~x = A ~ 0 = ~ 0. Therefore dim Null( B ) dim Null( AB ) , so p - dim Null( B ) p - dim Null( AB ) . Hence, rank( AB ) rank( B ), by the rank-nullity theorem. c) Prove that if

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## This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.

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A1_soln - Math 235 Assignment 1 Solutions 1. Let A be an m...

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