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Unformatted text preview: QUALIFYING EXAMINATION AUGUST 2001 MATH 554 - PROF. ARAPURA 1. ( 30 points ) Determine whether the following statements are true or false (you must justify the answer with either a proof or a counterexample). All matrices and vector spaces in this problem are defined over the field R . a) Let A and B be n n matrices. Then AB is invertible if and only if A and B are. b) Every square matrix is a product of elementary matrices. c) There exists a 3 2 matrix A and a 2 3 matrix B such that AB is invertible. d) If V and W are finite dimensional vector spaces such that dimV dimW , then V is isomorphic to a subspace of W . e) If v 1 ,v 2 ,v 3 are three distinct nonzero vectors in a finite dimensional vector space V , there exists a linear transformation f : V R satisfying f ( v i ) = i for all i . f) If v 1 ,v 2 ,v 3 are three linearly independent vectors in a finite dimensional vector space V , there exists a linear transformation f : V R satisfying f ( v i ) = i for all i...
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