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Unformatted text preview: [ UT ] . 13. Isomorphism is reexive: V V as witnessed by I V . Symmetric: if V W is witnessed by T , then T1 witnesses W V . Transitive: if V W and W Z , shown by T and U respectively, then V Z is shown by UT . 16. B1 ( cA + D ) B = cB1 AB + B1 DB by a few applications of Theorem 2.12, p. 89. Use exercise 6 twice to argue that if B1 AB = 0, A must be 0. Hence the null space is zero and is onetoone. Since the vector space is nitedimensional that suces to show is also onto and hence an isomorphism. 20. Since is an isomorphism, R ( L A ) = R ( L A ). Since is an isomorphism, by #17 R ( T ) has equal rank to R ( T ). Commutativity of Figure 2.2 then shows rank( T ) = rank( L A ). The nullity argument is similar. 1...
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 Winter '07
 Liu
 Linear Algebra, Algebra

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