Sample Final

Sample Final - MATH 225 SAMPLE FINAL EXAM PROBLEMS "1 $ $2...

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MATH 225 SAMPLE FINAL EXAM PROBLEMS 1. Can the matrix A exist or not exist: 1 2 1 1 1 2 1 1 1 1 1 1 1 1 1 0 0 0 1 1 0 0 0 1 2 " # $ $ $ $ $ $ % ' ' ' ' ' ' A 1 0 0 1 6 7 1 7 8 " # $ $ $ % ' ' ' = 1 0 1 2 1 1 3 0 1 4 1 1 5 1 2 " # $ $ $ $ $ $ % ' ' ' ' ' ' ? 2. Let U and V be subspaces of R 10 x 1 so that { α 1 , 2 , 3 , 4 , 5 , 6 } is a basis of U and { β 1 , 2 , 3 , 4 , 5 } is a basis of V . Argue that there is a nonzero γ U V . 3. Let T : V V be a linear transformation of the vector space V over R and set W = { V | for k 1 and depending on , T k ( ) = 0 V }. Show that W is a subspace of V . If T m ( ) = 0 but T m –1 ( ) 0 show that { , T ( ), T 2 ( ), . . . , T m –1 ( )} is linearly independent. 4. Determine if A = 3 0 ! 2 ! 2 1 2 1 0 0 " # $ $ % ' ' M 3 ( R ) is diagonalizable. If it is not, give reasons why. If A is diagonalizable find P M 3 ( R ) and a diagonal D M 3 ( R ) so that P –1 AP = D . 5. Let
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This note was uploaded on 12/16/2011 for the course MATH 225 at USC.

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