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Unformatted text preview: MAS 4105 Test 3 1. (8 pts) Indicate whether the following are true or false. i. If A,B M n n ( R ) and A is invertible, then det( AB ) = det( B ). T F ii. If A M n n ( R ) , then rank( A 2 ) < rank( A ). T F iii. If A M m n ( R ) , then rank( A ) m . T F iv. If A M n n ( R ) and 5 n then det( A ) = n i =1 ( 1) i +5 A i 5 det( A i 5 ). T F v. If A M m n ( R ) and n < m , then det( A ) = 0. T F vi. An elementary matrix is always invertible. T F vii. Every invertible matrix is the product of elementary matrices. T F viii. Let A M n n ( R ); the equation Ax = b is consistent if the reduced echelon form of ( A  b ) has n pivots. T F 2. (10 pts) Based on the following proofs, answer the questions below: Theorem 3.8. Let Ax = 0 be a homogeneous system of m linear equations in n unknowns over a field F . Let K denote the set of all solutions to Ax = 0. Then K = N ( L A ); hence K is a subspace of F n of dimension n rank( L A ) = n rank( A ). Proof. Clearly, K = { s F n : As = 0 } = N ( L A ) . The second part now follows from the dimension theorem. Corollary. If m < n , the system Ax = 0 has a nonzero solution. Proof. Suppose that m < n . Then rank( A ) = rank( L A ) m . Hence, dim( K ) = n rank( L A ) n m > , where K = N ( L A ) . Since dim( K ) > , K 6 = { } . Thus there exists a nonzero vector s K ; so s is a nonzero solution to Ax = 0...
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This note was uploaded on 12/27/2011 for the course MAS 4105 taught by Professor Rudyak during the Spring '09 term at University of Florida.
 Spring '09
 RUDYAK

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