MATH535_HW3 - Serge Ballif MATH 535 Homework 3 1 a Use what...

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Serge Ballif MATH 535 Homework 3 September 14, 2007 1. a) Use what you know about linear transformations to prove that dim Col ( AB ) dim Col ( A ) . b) Use what you know about linear transformations to prove that dim Col ( AB ) dim Col ( B ) . a) Let f A and f B denote the linear transformations defined by the matrices A and B . Then we have composition of maps l B -----→ f B m A -----→ f A n . The space Col ( AB ) is the image of the composite map f A f B . However, the image of the composite map is certainly contained in the image of the map f A = Col ( A ). Hence dim Col ( AB ) dim Col ( A ), with equality holding if f B is surjective. b) The dimension of the image of a linear transformation must be less than or equal to the dimension of the domain. Hence dim Col ( B ) = dim f B ( l ) dim f A f B ( l ) = dim Col ( AB ) , with equality holding if f A is injective. 2. a) Use that fact that rank( A t ) = rank( A ) and Problem 1 to show that dim Row ( AB ) min { dim Row ( A ) , dim Row ( B ) } .
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